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Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement

3.2.5  Cofinality using Axiom of Choice

Theoremalephreg 9401 A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
(cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)

Theorempwcfsdom 9402* A corollary of Konig's Theorem konigth 9388. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦)))       (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))

Theoremcfpwsdom 9403 A corollary of Konig's Theorem konigth 9388. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
𝐵 ∈ V       (2𝑜𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))

Theoremalephom 9404 From canth2 8110, we know that (ℵ‘0) < (2↑ω), but we cannot prove that (2↑ω) = (ℵ‘1) (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement (ℵ‘𝐴) < (2↑ω) is consistent for any ordinal 𝐴). However, we can prove that (2↑ω) is not equal to (ℵ‘ω), nor (ℵ‘(ℵ‘ω)), on cofinality grounds, because by Konig's Theorem konigth 9388 (in the form of cfpwsdom 9403), (2↑ω) has uncountable cofinality, which eliminates limit alephs like (ℵ‘ω). (The first limit aleph that is not eliminated is (ℵ‘(ℵ‘1)), which has cofinality (ℵ‘1).) (Contributed by Mario Carneiro, 21-Mar-2013.)
(card‘(2𝑜𝑚 ω)) ≠ (ℵ‘ω)

Theoremsmobeth 9405 The beth function is strictly monotone. This function is not strictly the beth function, but rather bethA is the same as (card‘(𝑅1‘(ω +𝑜 𝐴))), since conventionally we start counting at the first infinite level, and ignore the finite levels. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 2-Jun-2015.)
Smo (card ∘ 𝑅1)

3.3  ZFC Axioms with no distinct variable requirements

Theoremnd1 9406 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)

Theoremnd2 9407 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧𝑦)

Theoremnd3 9408 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)

Theoremnd4 9409 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦𝑥)

Theoremaxextnd 9410 A version of the Axiom of Extensionality with no distinct variable conditions. (Contributed by NM, 14-Aug-2003.)
𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)

Theoremaxrepndlem1 9411* Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))

Theoremaxrepndlem2 9412 Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
(((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))

Theoremaxrepnd 9413 A version of the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))

Theoremaxunndlem1 9414* Lemma for the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)

Theoremaxunnd 9415 A version of the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)

Theoremaxpowndlem1 9416 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))

Theoremaxpowndlem2 9417* Lemma for the Axiom of Power Sets with no distinct variable conditions. Revised to remove a redundant antecedent from the consequence. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Revised and shortened by Wolf Lammen, 9-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))

Theoremaxpowndlem3 9418* Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof shortened by Wolf Lammen, 10-Jun-2019.)
𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))

Theoremaxpowndlem4 9419 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
(¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))

Theoremaxpownd 9420 A version of the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))

Theoremaxregndlem1 9421 Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
(∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))

Theoremaxregndlem2 9422* Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))

Theoremaxregnd 9423 A version of the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))

Theoremaxinfndlem1 9424* Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)
(∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))

Theoremaxinfnd 9425 A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)
𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))

Theoremaxacndlem1 9426 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))

Theoremaxacndlem2 9427 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
(∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))

Theoremaxacndlem3 9428 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
(∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))

Theoremaxacndlem4 9429* Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Theoremaxacndlem5 9430* Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Theoremaxacnd 9431 A version of the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Theoremzfcndext 9432* Axiom of Extensionality ax-ext 2601, reproved from conditionless ZFC version and predicate calculus. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)

Theoremzfcndrep 9433* Axiom of Replacement ax-rep 4769, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
(∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))

Theoremzfcndun 9434* Axiom of Union ax-un 6946, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)

Theoremzfcndpow 9435* Axiom of Power Sets ax-pow 4841, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4855. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)

Theoremzfcndreg 9436* Axiom of Regularity ax-reg 8494, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
(∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))

Theoremzfcndinf 9437* Axiom of Infinity ax-inf 8532, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4845 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)
𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoremzfcndac 9438* Axiom of Choice ax-ac 9278, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))

3.4  The Generalized Continuum Hypothesis

3.4.1  Sets satisfying the Generalized Continuum Hypothesis

Syntaxcgch 9439 Extend class notation to include the collection of sets that satisfy the GCH.
class GCH

Definitiondf-gch 9440* Define the collection of "GCH-sets", or sets for which the generalized continuum hypothesis holds. In this language the generalized continuum hypothesis can be expressed as GCH = V. A set 𝑥 satisfies the generalized continuum hypothesis if it is finite or there is no set 𝑦 strictly between 𝑥 and its powerset in cardinality. The continuum hypothesis is equivalent to ω ∈ GCH. (Contributed by Mario Carneiro, 15-May-2015.)
GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥𝑦𝑦 ≺ 𝒫 𝑥)})

Theoremelgch 9441* Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
(𝐴𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))

Theoremfingch 9442 A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
Fin ⊆ GCH

Theoremgchi 9443 The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)

Theoremgchen1 9444 If 𝐴𝐵 < 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then 𝐴 = 𝐵 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
(((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)

Theoremgchen2 9445 If 𝐴 < 𝐵 ≤ 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then 𝐵 = 𝒫 𝐴 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
(((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≈ 𝒫 𝐴)

Theoremgchor 9446 If 𝐴𝐵 ≤ 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then either 𝐴 = 𝐵 or 𝐵 = 𝒫 𝐴 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
(((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → (𝐴𝐵𝐵 ≈ 𝒫 𝐴))

Theoremengch 9447 The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.)
(𝐴𝐵 → (𝐴 ∈ GCH ↔ 𝐵 ∈ GCH))

Theoremgchdomtri 9448 Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 9500. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴 ∈ GCH ∧ (𝐴 +𝑐 𝐴) ≈ 𝐴𝐵 ≼ 𝒫 𝐴) → (𝐴𝐵𝐵𝐴))

Theoremfpwwe2cbv 9449* Lemma for fpwwe2 9462. (Contributed by Mario Carneiro, 3-Jun-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}       𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}

Theoremfpwwe2lem1 9450* Lemma for fpwwe2 9462. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}       𝑊 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))

Theoremfpwwe2lem2 9451* Lemma for fpwwe2 9462. (Contributed by Mario Carneiro, 19-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)       (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))

Theoremfpwwe2lem3 9452* Lemma for fpwwe2 9462. (Contributed by Mario Carneiro, 19-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑋𝑊𝑅)       ((𝜑𝐵𝑋) → ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵)

Theoremfpwwe2lem5 9453* Lemma for fpwwe2 9462. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)       ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)

Theoremfpwwe2lem6 9454* Lemma for fpwwe2 9462. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑𝐵 ∈ dom 𝑀)    &   (𝜑𝐵 ∈ dom 𝑁)    &   (𝜑 → (𝑀𝐵) = (𝑁𝐵))       ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑋𝐶𝑌 ∧ (𝑀𝐶) = (𝑁𝐶)))

Theoremfpwwe2lem7 9455* Lemma for fpwwe2 9462. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑𝐵 ∈ dom 𝑀)    &   (𝜑𝐵 ∈ dom 𝑁)    &   (𝜑 → (𝑀𝐵) = (𝑁𝐵))       ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑆(𝑁𝐵) ∧ (𝐷𝑅(𝑀𝐵) → (𝐶𝑅𝐷𝐶𝑆𝐷))))

Theoremfpwwe2lem8 9456* Lemma for fpwwe2 9462. Show by induction that the two isometries 𝑀 and 𝑁 agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑 → dom 𝑀 ⊆ dom 𝑁)       (𝜑𝑀 = (𝑁 ↾ dom 𝑀))

Theoremfpwwe2lem9 9457* Lemma for fpwwe2 9462. Given two well-orders 𝑋, 𝑅 and 𝑌, 𝑆 of parts of 𝐴, one is an initial segment of the other. (The 𝑂𝑃 hypothesis is in order to break the symmetry of 𝑋 and 𝑌.) (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑 → dom 𝑀 ⊆ dom 𝑁)       (𝜑 → (𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))

Theoremfpwwe2lem10 9458* Lemma for fpwwe2 9462. Given two well-orders 𝑋, 𝑅 and 𝑌, 𝑆 of parts of 𝐴, one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)       (𝜑 → ((𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))

Theoremfpwwe2lem11 9459* Lemma for fpwwe2 9462. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))

Theoremfpwwe2lem12 9460* Lemma for fpwwe2 9462. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑𝑋 ∈ dom 𝑊)

Theoremfpwwe2lem13 9461* Lemma for fpwwe2 9462. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑 → (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)

Theoremfpwwe2 9462* Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset 𝑋, (𝑊𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 8850. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))

Theoremfpwwecbv 9463* Lemma for fpwwe 9465. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}       𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))}

Theoremfpwwelem 9464* Lemma for fpwwe 9465. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}    &   (𝜑𝐴 ∈ V)       (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))))

Theoremfpwwe 9465* Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 8110 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset 𝑋, (𝑊𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 8850. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))

Theoremcanth4 9466* An "effective" form of Cantor's theorem canth 6605. For any function 𝐹 from the powerset of 𝐴 to 𝐴, there are two definable sets 𝐵 and 𝐶 which witness non-injectivity of 𝐹. Corollary 1.3 of [KanamoriPincus] p. 416. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}    &   𝐵 = dom 𝑊    &   𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})       ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))

Theoremcanthnumlem 9467* Lemma for canthnum 9468. (Contributed by Mario Carneiro, 19-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}    &   𝐵 = dom 𝑊    &   𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})       (𝐴𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)

Theoremcanthnum 9468 The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8110. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)
(𝐴𝑉𝐴 ≺ (𝒫 𝐴 ∩ dom card))

Theoremcanthwelem 9469* Lemma for canthwe 9470. (Contributed by Mario Carneiro, 31-May-2015.)
𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}    &   𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   𝐵 = dom 𝑊    &   𝐶 = ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})       (𝐴𝑉 → ¬ 𝐹:𝑂1-1𝐴)

Theoremcanthwe 9470* The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8110. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)
𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}       (𝐴𝑉𝐴𝑂)

Theoremcanthp1lem1 9471 Lemma for canthp1 9473. (Contributed by Mario Carneiro, 18-May-2015.)
(1𝑜𝐴 → (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴)

Theoremcanthp1lem2 9472* Lemma for canthp1 9473. (Contributed by Mario Carneiro, 18-May-2015.)
(𝜑 → 1𝑜𝐴)    &   (𝜑𝐹:𝒫 𝐴1-1-onto→(𝐴 +𝑐 1𝑜))    &   (𝜑𝐺:((𝐴 +𝑐 1𝑜) ∖ {(𝐹𝐴)})–1-1-onto𝐴)    &   𝐻 = ((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))    &   𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐻‘(𝑟 “ {𝑦})) = 𝑦))}    &   𝐵 = dom 𝑊        ¬ 𝜑

Theoremcanthp1 9473 A slightly stronger form of Cantor's theorem: For 1 < 𝑛, 𝑛 + 1 < 2↑𝑛. Corollary 1.6 of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 18-May-2015.)
(1𝑜𝐴 → (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴)

Theoremfinngch 9474 The exclusion of finite sets from consideration in df-gch 9440 is necessary, because otherwise finite sets larger than a singleton would violate the GCH property. (Contributed by Mario Carneiro, 10-Jun-2015.)
((𝐴 ∈ Fin ∧ 1𝑜𝐴) → (𝐴 ≺ (𝐴 +𝑐 1𝑜) ∧ (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴))

Theoremgchcda1 9475 An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.)
((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 +𝑐 1𝑜) ≈ 𝐴)

Theoremgchinf 9476 An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝐴)

Theorempwfseqlem1 9477* Lemma for pwfseq 9483. Derive a contradiction by diagonalization. (Contributed by Mario Carneiro, 31-May-2015.)
(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)    &   𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})       ((𝜑𝜓) → 𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)))

Theorempwfseqlem2 9478* Lemma for pwfseq 9483. (Contributed by Mario Carneiro, 18-Nov-2014.) (Revised by AV, 18-Sep-2021.)
(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)    &   𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})    &   𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))       ((𝑌 ∈ Fin ∧ 𝑅𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))

Theorempwfseqlem3 9479* Lemma for pwfseq 9483. Using the construction 𝐷 from pwfseqlem1 9477, produce a function 𝐹 that maps any well-ordered infinite set to an element outside the set. (Contributed by Mario Carneiro, 31-May-2015.)
(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)    &   𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})    &   𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))       ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))

Theorempwfseqlem4a 9480* Lemma for pwfseqlem4 9481. (Contributed by Mario Carneiro, 7-Jun-2016.)
(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)    &   𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})    &   𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))       ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)

Theorempwfseqlem4 9481* Lemma for pwfseq 9483. Derive a final contradiction from the function 𝐹 in pwfseqlem3 9479. Applying fpwwe2 9462 to it, we get a certain maximal well-ordered subset 𝑍, but the defining property (𝑍𝐹(𝑊𝑍)) ∈ 𝑍 contradicts our assumption on 𝐹, so we are reduced to the case of 𝑍 finite. This too is a contradiction, though, because 𝑍 and its preimage under (𝑊𝑍) are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)
(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)    &   𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})    &   𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))    &   𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏𝑎 [(𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))}    &   𝑍 = dom 𝑊        ¬ 𝜑

Theorempwfseqlem5 9482* Lemma for pwfseq 9483. Although in some ways pwfseqlem4 9481 is the "main" part of the proof, one last aspect which makes up a remark in the original text is by far the hardest part to formalize. The main proof relies on the existence of an injection 𝐾 from the set of finite sequences on an infinite set 𝑥 to 𝑥. Now this alone would not be difficult to prove; this is mostly the claim of fseqen 8847. However, what is needed for the proof is a canonical injection on these sets, so we have to start from scratch pulling together explicit bijections from the lemmas.

If one attempts such a program, it will mostly go through, but there is one key step which is inherently nonconstructive, namely the proof of infxpen 8834. The resolution is not obvious, but it turns out that reversing an infinite ordinal's Cantor normal form absorbs all the non-leading terms (cnfcom3c 8600), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 8838). (Contributed by Mario Carneiro, 31-May-2015.)

(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))    &   (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))    &   𝑂 = OrdIso(𝑟, 𝑡)    &   𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩)    &   𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇)    &   𝑆 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡𝑚 suc 𝑘) ↦ ((𝑓‘(𝑥𝑘))𝑃(𝑥𝑘)))), {⟨∅, (𝑂‘∅)⟩})    &   𝑄 = (𝑦 𝑛 ∈ ω (𝑡𝑚 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)    &   𝐼 = (𝑥 ∈ ω, 𝑦𝑡 ↦ ⟨(𝑂𝑥), 𝑦⟩)    &   𝐾 = ((𝑃𝐼) ∘ 𝑄)        ¬ 𝜑

Theorempwfseq 9483* The powerset of a Dedekind-infinite set does not inject into the set of finite sequences. The proof is due to Halbeisen and Shelah. Proposition 1.7 of [KanamoriPincus] p. 418. (Contributed by Mario Carneiro, 31-May-2015.)
(ω ≼ 𝐴 → ¬ 𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))

Theorempwxpndom2 9484 The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
(ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 (𝐴 × 𝐴)))

Theorempwxpndom 9485 The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
(ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 × 𝐴))

Theorempwcdandom 9486 The powerset of a Dedekind-infinite set does not inject into its cardinal sum with itself. (Contributed by Mario Carneiro, 31-May-2015.)
(ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 +𝑐 𝐴))

Theoremgchcdaidm 9487 An infinite GCH-set is idempotent under cardinal sum. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 +𝑐 𝐴) ≈ 𝐴)

Theoremgchxpidm 9488 An infinite GCH-set is idempotent under cardinal product. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 × 𝐴) ≈ 𝐴)

Theoremgchpwdom 9489 A relationship between dominance over the powerset and strict dominance when the sets involved are infinite GCH-sets. Proposition 3.1 of [KanamoriPincus] p. 421. (Contributed by Mario Carneiro, 31-May-2015.)
((ω ≼ 𝐴𝐴 ∈ GCH ∧ 𝐵 ∈ GCH) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))

Theoremgchaleph 9490 If (ℵ‘𝐴) is a GCH-set and its powerset is well-orderable, then the successor aleph (ℵ‘suc 𝐴) is equinumerous to the powerset of (ℵ‘𝐴). (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))

Theoremgchaleph2 9491 If (ℵ‘𝐴) and (ℵ‘suc 𝐴) are GCH-sets, then the successor aleph (ℵ‘suc 𝐴) is equinumerous to the powerset of (ℵ‘𝐴). (Contributed by Mario Carneiro, 31-May-2015.)
((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))

Theoremhargch 9492 If 𝐴 + ≈ 𝒫 𝐴, then 𝐴 is a GCH-set. The much simpler converse to gchhar 9498. (Contributed by Mario Carneiro, 2-Jun-2015.)
((har‘𝐴) ≈ 𝒫 𝐴𝐴 ∈ GCH)

Theoremalephgch 9493 If (ℵ‘suc 𝐴) is equinumerous to the powerset of (ℵ‘𝐴), then (ℵ‘𝐴) is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
((ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴) → (ℵ‘𝐴) ∈ GCH)

Theoremgch2 9494 It is sufficient to require that all alephs are GCH-sets to ensure the full generalized continuum hypothesis. (The proof uses the Axiom of Regularity.) (Contributed by Mario Carneiro, 15-May-2015.)
(GCH = V ↔ ran ℵ ⊆ GCH)

Theoremgch3 9495 An equivalent formulation of the generalized continuum hypothesis. (Contributed by Mario Carneiro, 15-May-2015.)
(GCH = V ↔ ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))

Theoremgch-kn 9496* The equivalence of two versions of the Generalized Continuum Hypothesis. The right-hand side is the standard version in the literature. The left-hand side is a version devised by Kannan Nambiar, which he calls the Axiom of Combinatorial Sets. For the notation and motivation behind this axiom, see his paper, "Derivation of Continuum Hypothesis from Axiom of Combinatorial Sets," available at http://www.e-atheneum.net/science/derivation_ch.pdf. The equivalence of the two sides provides a negative answer to Open Problem 2 in http://www.e-atheneum.net/science/open_problem_print.pdf. The key idea in the proof below is to equate both sides of alephexp2 9400 to the successor aleph using enen2 8098. (Contributed by NM, 1-Oct-2004.)
(𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (ℵ‘suc 𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))))

3.4.2  Derivation of the Axiom of Choice

Theoremgchaclem 9497 Lemma for gchac 9500 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑 → ω ≼ 𝐴)    &   (𝜑 → 𝒫 𝐶 ∈ GCH)    &   (𝜑 → (𝐴𝐶 ∧ (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴𝐵)))       (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴𝐵)))

Theoremgchhar 9498 A "local" form of gchac 9500. If 𝐴 and 𝒫 𝐴 are GCH-sets, then the Hartogs number of 𝐴 is 𝒫 𝐴 (so 𝒫 𝐴 and a fortiori 𝐴 are well-orderable). The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
((ω ≼ 𝐴𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≈ 𝒫 𝐴)

Theoremgchacg 9499 A "local" form of gchac 9500. If 𝐴 and 𝒫 𝐴 are GCH-sets, then 𝒫 𝐴 is well-orderable. The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 15-May-2015.)
((ω ≼ 𝐴𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ∈ dom card)

Theoremgchac 9500 The Generalized Continuum Hypothesis implies the Axiom of Choice. The original proof is due to Sierpiński (1947); we use a refinement of Sierpiński's result due to Specker. (Contributed by Mario Carneiro, 15-May-2015.)
(GCH = V → CHOICE)

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