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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | axregnd 10601 | A version of the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) (New usage is discouraged.) |
⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) | ||
Theorem | axinfndlem1 10602* | Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.) |
⊢ (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) | ||
Theorem | axinfnd 10603 | A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.) |
⊢ ∃𝑥(𝑦 ∈ 𝑧 → (𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) | ||
Theorem | axacndlem1 10604 | Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))) | ||
Theorem | axacndlem2 10605 | Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))) | ||
Theorem | axacndlem3 10606 | Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑦 𝑦 = 𝑧 → ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤))) | ||
Theorem | axacndlem4 10607* | 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.) |
⊢ ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)) | ||
Theorem | axacndlem5 10608* | 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.) |
⊢ ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)) | ||
Theorem | axacnd 10609 | 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.) |
⊢ ∃𝑥∀𝑦∀𝑧(∀𝑥(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) → ∃𝑤∀𝑦(∃𝑤((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑤) ∧ (𝑦 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) ↔ 𝑦 = 𝑤)) | ||
Theorem | zfcndext 10610* | Axiom of Extensionality ax-ext 2701, reproved from conditionless ZFC version and predicate calculus. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | ||
Theorem | zfcndrep 10611* | Axiom of Replacement ax-rep 5284, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) | ||
Theorem | zfcndun 10612* | Axiom of Union ax-un 7727, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
Theorem | zfcndpow 10613* | Axiom of Power Sets ax-pow 5362, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness" dtru 5435. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
Theorem | zfcndreg 10614* | Axiom of Regularity ax-reg 9589, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) | ||
Theorem | zfcndinf 10615* | Axiom of Infinity ax-inf 9635, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing Theorem el 5436 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.) |
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) | ||
Theorem | zfcndac 10616* | Axiom of Choice ax-ac 10456, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) | ||
Syntax | cgch 10617 | Extend class notation to include the collection of sets that satisfy the GCH. |
class GCH | ||
Definition | df-gch 10618* | 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 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)}) | ||
Theorem | elgch 10619* | Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴)))) | ||
Theorem | fingch 10620 | A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ Fin ⊆ GCH | ||
Theorem | gchi 10621 | 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 ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin) | ||
Theorem | gchen1 10622 | If 𝐴 ≤ 𝐵 < 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) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴)) → 𝐴 ≈ 𝐵) | ||
Theorem | gchen2 10623 | 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) ∧ (𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≈ 𝒫 𝐴) | ||
Theorem | gchor 10624 | If 𝐴 ≤ 𝐵 ≤ 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) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴)) → (𝐴 ≈ 𝐵 ∨ 𝐵 ≈ 𝒫 𝐴)) | ||
Theorem | engch 10625 | The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ GCH ↔ 𝐵 ∈ GCH)) | ||
Theorem | gchdomtri 10626 | Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 10678. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝐴 ∈ GCH ∧ (𝐴 ⊔ 𝐴) ≈ 𝐴 ∧ 𝐵 ≼ 𝒫 𝐴) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)) | ||
Theorem | fpwwe2cbv 10627* | Lemma for fpwwe2 10640. (Contributed by Mario Carneiro, 3-Jun-2015.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} ⇒ ⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))} | ||
Theorem | fpwwe2lem1 10628* | Lemma for fpwwe2 10640. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} ⇒ ⊢ 𝑊 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) | ||
Theorem | fpwwe2lem2 10629* | Lemma for fpwwe2 10640. (Contributed by Mario Carneiro, 19-May-2015.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))) | ||
Theorem | fpwwe2lem3 10630* | Lemma for fpwwe2 10640. (Contributed by Mario Carneiro, 19-May-2015.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋𝑊𝑅) ⇒ ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵) | ||
Theorem | fpwwe2lem4 10631* | Lemma for fpwwe2 10640. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) ⇒ ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴) | ||
Theorem | fpwwe2lem5 10632* | Lemma for fpwwe2 10640. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) & ⊢ (𝜑 → 𝑋𝑊𝑅) & ⊢ (𝜑 → 𝑌𝑊𝑆) & ⊢ 𝑀 = OrdIso(𝑅, 𝑋) & ⊢ 𝑁 = OrdIso(𝑆, 𝑌) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑁) & ⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵)) ⇒ ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ (◡𝑀‘𝐶) = (◡𝑁‘𝐶))) | ||
Theorem | fpwwe2lem6 10633* | Lemma for fpwwe2 10640. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) & ⊢ (𝜑 → 𝑋𝑊𝑅) & ⊢ (𝜑 → 𝑌𝑊𝑆) & ⊢ 𝑀 = OrdIso(𝑅, 𝑋) & ⊢ 𝑁 = OrdIso(𝑆, 𝑌) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑁) & ⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵)) ⇒ ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶𝑆(𝑁‘𝐵) ∧ (𝐷𝑅(𝑀‘𝐵) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷)))) | ||
Theorem | fpwwe2lem7 10634* | Lemma for fpwwe2 10640. Show by induction that the two isometries 𝑀 and 𝑁 agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) & ⊢ (𝜑 → 𝑋𝑊𝑅) & ⊢ (𝜑 → 𝑌𝑊𝑆) & ⊢ 𝑀 = OrdIso(𝑅, 𝑋) & ⊢ 𝑁 = OrdIso(𝑆, 𝑌) & ⊢ (𝜑 → dom 𝑀 ⊆ dom 𝑁) ⇒ ⊢ (𝜑 → 𝑀 = (𝑁 ↾ dom 𝑀)) | ||
Theorem | fpwwe2lem8 10635* | Lemma for fpwwe2 10640. Given two well-orders ⟨𝑋, 𝑅⟩ and ⟨𝑌, 𝑆⟩ of parts of 𝐴, one is an initial segment of the other. (The 𝑂 ⊆ 𝑃 hypothesis is in order to break the symmetry of 𝑋 and 𝑌.) (Contributed by Mario Carneiro, 15-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) & ⊢ (𝜑 → 𝑋𝑊𝑅) & ⊢ (𝜑 → 𝑌𝑊𝑆) & ⊢ 𝑀 = OrdIso(𝑅, 𝑋) & ⊢ 𝑁 = OrdIso(𝑆, 𝑌) & ⊢ (𝜑 → dom 𝑀 ⊆ dom 𝑁) ⇒ ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋)))) | ||
Theorem | fpwwe2lem9 10636* | Lemma for fpwwe2 10640. 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.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) & ⊢ (𝜑 → 𝑋𝑊𝑅) & ⊢ (𝜑 → 𝑌𝑊𝑆) ⇒ ⊢ (𝜑 → ((𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌))))) | ||
Theorem | fpwwe2lem10 10637* | Lemma for fpwwe2 10640. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) & ⊢ 𝑋 = ∪ dom 𝑊 ⇒ ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋)) | ||
Theorem | fpwwe2lem11 10638* | Lemma for fpwwe2 10640. (Contributed by Mario Carneiro, 18-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) & ⊢ 𝑋 = ∪ dom 𝑊 ⇒ ⊢ (𝜑 → 𝑋 ∈ dom 𝑊) | ||
Theorem | fpwwe2lem12 10639* | Lemma for fpwwe2 10640. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) & ⊢ 𝑋 = ∪ dom 𝑊 ⇒ ⊢ (𝜑 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) | ||
Theorem | fpwwe2 10640* | Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset ⟨𝑋, (𝑊‘𝑋)⟩ 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 10027. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) & ⊢ 𝑋 = ∪ dom 𝑊 ⇒ ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)))) | ||
Theorem | fpwwecbv 10641* | Lemma for fpwwe 10643. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))} ⇒ ⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (𝐹‘(◡𝑠 “ {𝑧})) = 𝑧))} | ||
Theorem | fpwwelem 10642* | Lemma for fpwwe 10643. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦)))) | ||
Theorem | fpwwe 10643* | Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 9132 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset ⟨𝑋, (𝑊‘𝑋)⟩ 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 10027. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))} & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹‘𝑥) ∈ 𝐴) & ⊢ 𝑋 = ∪ dom 𝑊 ⇒ ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹‘𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)))) | ||
Theorem | canth4 10644* | An "effective" form of Cantor's theorem canth 7364. 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⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))} & ⊢ 𝐵 = ∪ dom 𝑊 & ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶))) | ||
Theorem | canthnumlem 10645* | Lemma for canthnum 10646. (Contributed by Mario Carneiro, 19-May-2015.) |
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))} & ⊢ 𝐵 = ∪ dom 𝑊 & ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) ⇒ ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) | ||
Theorem | canthnum 10646 | The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9132. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ (𝒫 𝐴 ∩ dom card)) | ||
Theorem | canthwelem 10647* | Lemma for canthwe 10648. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} & ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} & ⊢ 𝐵 = ∪ dom 𝑊 & ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ⇒ ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:𝑂–1-1→𝐴) | ||
Theorem | canthwe 10648* | The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9132. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝑂) | ||
Theorem | canthp1lem1 10649 | Lemma for canthp1 10651. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 2o) ≼ 𝒫 𝐴) | ||
Theorem | canthp1lem2 10650* | Lemma for canthp1 10651. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝜑 → 1o ≺ 𝐴) & ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o)) & ⊢ (𝜑 → 𝐺:((𝐴 ⊔ 1o) ∖ {(𝐹‘𝐴)})–1-1-onto→𝐴) & ⊢ 𝐻 = ((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥))) & ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐻‘(◡𝑟 “ {𝑦})) = 𝑦))} & ⊢ 𝐵 = ∪ dom 𝑊 ⇒ ⊢ ¬ 𝜑 | ||
Theorem | canthp1 10651 | 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.) |
⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≺ 𝒫 𝐴) | ||
Theorem | finngch 10652 | The exclusion of finite sets from consideration in df-gch 10618 is necessary, because otherwise finite sets larger than a singleton would violate the GCH property. (Contributed by Mario Carneiro, 10-Jun-2015.) |
⊢ ((𝐴 ∈ Fin ∧ 1o ≺ 𝐴) → (𝐴 ≺ (𝐴 ⊔ 1o) ∧ (𝐴 ⊔ 1o) ≺ 𝒫 𝐴)) | ||
Theorem | gchdju1 10653 | An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ⊔ 1o) ≈ 𝐴) | ||
Theorem | gchinf 10654 | An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝐴) | ||
Theorem | pwfseqlem1 10655* | Lemma for pwfseq 10661. Derive a contradiction by diagonalization. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) & ⊢ (𝜑 → 𝑋 ⊆ 𝐴) & ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋) & ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥) & ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ (∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∖ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛))) | ||
Theorem | pwfseqlem2 10656* | Lemma for pwfseq 10661. (Contributed by Mario Carneiro, 18-Nov-2014.) (Revised by AV, 18-Sep-2021.) |
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) & ⊢ (𝜑 → 𝑋 ⊆ 𝐴) & ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋) & ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥) & ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) & ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))) ⇒ ⊢ ((𝑌 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌))) | ||
Theorem | pwfseqlem3 10657* | Lemma for pwfseq 10661. Using the construction 𝐷 from pwfseqlem1 10655, produce a function 𝐹 that maps any well-ordered infinite set to an element outside the set. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) & ⊢ (𝜑 → 𝑋 ⊆ 𝐴) & ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋) & ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥) & ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) & ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥)) | ||
Theorem | pwfseqlem4a 10658* | Lemma for pwfseqlem4 10659. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) & ⊢ (𝜑 → 𝑋 ⊆ 𝐴) & ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋) & ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥) & ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) & ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))) ⇒ ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴) | ||
Theorem | pwfseqlem4 10659* | Lemma for pwfseq 10661. Derive a final contradiction from the function 𝐹 in pwfseqlem3 10657. Applying fpwwe2 10640 to it, we get a certain maximal well-ordered subset 𝑍, but the defining property (𝑍𝐹(𝑊‘𝑍)) ∈ 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 (𝑊‘𝑍) 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.) |
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) & ⊢ (𝜑 → 𝑋 ⊆ 𝐴) & ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋) & ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥) & ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))}) & ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))) & ⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏 ∈ 𝑎 [(◡𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))} & ⊢ 𝑍 = ∪ dom 𝑊 ⇒ ⊢ ¬ 𝜑 | ||
Theorem | pwfseqlem5 10660* |
Lemma for pwfseq 10661. Although in some ways pwfseqlem4 10659 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 10024. 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 10011. 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 9703), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 10015). (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) & ⊢ (𝜑 → 𝑋 ⊆ 𝐴) & ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋) & ⊢ (𝜓 ↔ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡)) & ⊢ (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) & ⊢ 𝑂 = OrdIso(𝑟, 𝑡) & ⊢ 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩) & ⊢ 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ◡𝑇) & ⊢ 𝑆 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡 ↑m suc 𝑘) ↦ ((𝑓‘(𝑥 ↾ 𝑘))𝑃(𝑥‘𝑘)))), {⟨∅, (𝑂‘∅)⟩}) & ⊢ 𝑄 = (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑡 ↑m 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩) & ⊢ 𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩) & ⊢ 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | pwfseq 10661* | 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 (𝐴 ↑m 𝑛)) | ||
Theorem | pwxpndom2 10662 | The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) (Proof shortened by AV, 18-Jul-2022.) |
⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | ||
Theorem | pwxpndom 10663 | 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)) | ||
Theorem | pwdjundom 10664 | 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 ⊔ 𝐴)) | ||
Theorem | gchdjuidm 10665 | 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 ⊔ 𝐴) ≈ 𝐴) | ||
Theorem | gchxpidm 10666 | 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) | ||
Theorem | gchpwdom 10667 | 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 ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝐵 ∈ GCH) → (𝐴 ≺ 𝐵 ↔ 𝒫 𝐴 ≼ 𝐵)) | ||
Theorem | gchaleph 10668 | 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)) | ||
Theorem | gchaleph2 10669 | 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)) | ||
Theorem | hargch 10670 | If 𝐴 + ≈ 𝒫 𝐴, then 𝐴 is a GCH-set. The much simpler converse to gchhar 10676. (Contributed by Mario Carneiro, 2-Jun-2015.) |
⊢ ((har‘𝐴) ≈ 𝒫 𝐴 → 𝐴 ∈ GCH) | ||
Theorem | alephgch 10671 | 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) | ||
Theorem | gch2 10672 | 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) | ||
Theorem | gch3 10673 | 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)) | ||
Theorem | gch-kn 10674* | 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 10578 to the successor aleph using enen2 9120. (Contributed by NM, 1-Oct-2004.) |
⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (ℵ‘suc 𝐴) ≈ (2o ↑m (ℵ‘𝐴)))) | ||
Theorem | gchaclem 10675 | Lemma for gchac 10678 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → ω ≼ 𝐴) & ⊢ (𝜑 → 𝒫 𝐶 ∈ GCH) & ⊢ (𝜑 → (𝐴 ≼ 𝐶 ∧ (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) ⇒ ⊢ (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) | ||
Theorem | gchhar 10676 | A "local" form of gchac 10678. 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 ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≈ 𝒫 𝐴) | ||
Theorem | gchacg 10677 | A "local" form of gchac 10678. 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 ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ∈ dom card) | ||
Theorem | gchac 10678 | 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) | ||
Here we introduce Tarski-Grothendieck (TG) set theory, named after mathematicians Alfred Tarski and Alexander Grothendieck. TG theory extends ZFC with the TG Axiom ax-groth 10820, which states that for every set 𝑥 there is an inaccessible cardinal 𝑦 such that 𝑦 is not in 𝑥. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics". We first introduce the concept of inaccessibles, including weakly and strongly inaccessible cardinals (df-wina 10681 and df-ina 10682 respectively ), Tarski classes (df-tsk 10746), and Grothendieck universes (df-gru 10788). We then introduce the Tarski's axiom ax-groth 10820 and prove various properties from that. | ||
Syntax | cwina 10679 | The class of weak inaccessibles. |
class Inaccw | ||
Syntax | cina 10680 | The class of strong inaccessibles. |
class Inacc | ||
Definition | df-wina 10681* | An ordinal is weakly inaccessible iff it is a regular limit cardinal. Note that our definition allows ω as a weakly inaccessible cardinal. (Contributed by Mario Carneiro, 22-Jun-2013.) |
⊢ Inaccw = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑥 𝑦 ≺ 𝑧)} | ||
Definition | df-ina 10682* | An ordinal is strongly inaccessible iff it is a regular strong limit cardinal, which is to say that it dominates the powersets of every smaller ordinal. (Contributed by Mario Carneiro, 22-Jun-2013.) |
⊢ Inacc = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ≺ 𝑥)} | ||
Theorem | elwina 10683* | Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.) |
⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | ||
Theorem | elina 10684* | Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.) |
⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)) | ||
Theorem | winaon 10685 | A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On) | ||
Theorem | inawinalem 10686* | Lemma for inawina 10687. (Contributed by Mario Carneiro, 8-Jun-2014.) |
⊢ (𝐴 ∈ On → (∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | ||
Theorem | inawina 10687 | Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw) | ||
Theorem | omina 10688 | ω is a strongly inaccessible cardinal. (Many definitions of "inaccessible" explicitly disallow ω as an inaccessible cardinal, but this choice allows to reuse our results for inaccessibles for ω.) (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ ω ∈ Inacc | ||
Theorem | winacard 10689 | A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴) | ||
Theorem | winainflem 10690* | A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → ω ⊆ 𝐴) | ||
Theorem | winainf 10691 | A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴) | ||
Theorem | winalim 10692 | A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ (𝐴 ∈ Inaccw → Lim 𝐴) | ||
Theorem | winalim2 10693* | A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) | ||
Theorem | winafp 10694 | A nontrivial weakly inaccessible cardinal is a fixed point of the aleph function. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (ℵ‘𝐴) = 𝐴) | ||
Theorem | winafpi 10695 | This theorem, which states that a nontrivial inaccessible cardinal is its own aleph number, is stated here in inference form, where the assumptions are in the hypotheses rather than an antecedent. Often, we use dedth 4585 to turn this type of statement into the closed form statement winafp 10694, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 10694 using this theorem and dedth 4585, in ZFC. (You can prove this if you use ax-groth 10820, though.) (Contributed by Mario Carneiro, 28-May-2014.) |
⊢ 𝐴 ∈ Inaccw & ⊢ 𝐴 ≠ ω ⇒ ⊢ (ℵ‘𝐴) = 𝐴 | ||
Theorem | gchina 10696 | Assuming the GCH, weakly and strongly inaccessible cardinals coincide. Theorem 11.20 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 5-Jun-2015.) |
⊢ (GCH = V → Inaccw = Inacc) | ||
Syntax | cwun 10697 | Extend class definition to include the class of all weak universes. |
class WUni | ||
Syntax | cwunm 10698 | Extend class definition to include the map whose value is the smallest weak universe of which the given set is a subset. |
class wUniCl | ||
Definition | df-wun 10699* | The class of all weak universes. A weak universe is a nonempty transitive class closed under union, pairing, and powerset. The advantage of weak universes over Grothendieck universes is that one can prove that every set is contained in a weak universe in ZF (see uniwun 10737) whereas the analogue for Grothendieck universes requires ax-groth 10820 (see grothtsk 10832). (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))} | ||
Definition | df-wunc 10700* | A function that maps a set 𝑥 to the smallest weak universe that contains the elements of the set. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ wUniCl = (𝑥 ∈ V ↦ ∩ {𝑢 ∈ WUni ∣ 𝑥 ⊆ 𝑢}) |
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