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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | infmap 10001* | An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. (Contributed by NM, 1-Oct-2004.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) |
⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) | ||
Theorem | alephadd 10002 | The sum of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ ((ℵ‘𝐴) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) | ||
Theorem | alephmul 10003 | The product of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) × (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))) | ||
Theorem | alephexp1 10004 | An exponentiation law for alephs. Lemma 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ((ℵ‘𝐴) ↑m (ℵ‘𝐵)) ≈ (2o ↑m (ℵ‘𝐵))) | ||
Theorem | alephsuc3 10005* | An alternate representation of a successor aleph. Compare alephsuc 9497 and alephsuc2 9509. Equality can be obtained by taking the card of the right-hand side then using alephcard 9499 and carden 9976. (Contributed by NM, 23-Oct-2004.) |
⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}) | ||
Theorem | alephexp2 10006* | An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 10004 (which works if the base is less than or equal to the exponent) and infmap 10001 (which works if the exponent is less than or equal to the base). They can be equated only when the base is equal to the exponent, and this is the result. (Contributed by NM, 23-Oct-2004.) |
⊢ (𝐴 ∈ On → (2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}) | ||
Theorem | alephreg 10007 | 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) | ||
Theorem | pwcfsdom 10008* | A corollary of Konig's Theorem konigth 9994. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.) |
⊢ 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦))) ⇒ ⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))) | ||
Theorem | cfpwsdom 10009 | A corollary of Konig's Theorem konigth 9994. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) | ||
Theorem | alephom 10010 | From canth2 8673, 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 9994 (in the form of cfpwsdom 10009), (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‘(2o ↑m ω)) ≠ (ℵ‘ω) | ||
Theorem | smobeth 10011 | 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) | ||
Theorem | nd1 10012 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧) | ||
Theorem | nd2 10013 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) | ||
Theorem | nd3 10014 | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦) | ||
Theorem | nd4 10015 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥) | ||
Theorem | axextnd 10016 | A version of the Axiom of Extensionality with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 14-Aug-2003.) (New usage is discouraged.) |
⊢ ∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) | ||
Theorem | axrepndlem1 10017* | Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))) | ||
Theorem | axrepndlem2 10018 | Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (New usage is discouraged.) |
⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))) | ||
Theorem | axrepnd 10019 | A version of the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) | ||
Theorem | axunndlem1 10020* | Lemma for the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
Theorem | axunnd 10021 | A version of the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
Theorem | axpowndlem1 10022 | Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) | ||
Theorem | axpowndlem2 10023* | Lemma for the Axiom of Power Sets with no distinct variable conditions. Revised to remove a redundant antecedent from the consequence. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Revised and shortened by Wolf Lammen, 9-Jun-2019.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) | ||
Theorem | axpowndlem3 10024* | Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof shortened by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.) |
⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) | ||
Theorem | axpowndlem4 10025 | Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.) |
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))) | ||
Theorem | axpownd 10026 | A version of the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 4-Jan-2002.) (New usage is discouraged.) |
⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) | ||
Theorem | axregndlem1 10027 | Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))) | ||
Theorem | axregndlem2 10028* | Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.) |
⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) | ||
Theorem | axregnd 10029 | A version of the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (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 10030* | 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 10031 | 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 10032 | Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (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 10033 | Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (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 10034 | Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2389. (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 10035* | 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 10036* | 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 10037 | 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 10038* | Axiom of Extensionality ax-ext 2796, reproved from conditionless ZFC version and predicate calculus. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | ||
Theorem | zfcndrep 10039* | Axiom of Replacement ax-rep 5193, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2389. (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 10040* | Axiom of Union ax-un 7464, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2389. (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 10041* | Axiom of Power Sets ax-pow 5269, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness" dtru 5274. Usage of this theorem is discouraged because it depends on ax-13 2389. (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 10042* | Axiom of Regularity ax-reg 9059, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2389. (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 10043* | Axiom of Infinity ax-inf 9104, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 5273 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 10044* | Axiom of Choice ax-ac 9884, 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 10045 | Extend class notation to include the collection of sets that satisfy the GCH. |
class GCH | ||
Definition | df-gch 10046* | 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 10047* | 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 10048 | A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ Fin ⊆ GCH | ||
Theorem | gchi 10049 | 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 10050 | 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 10051 | 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 10052 | 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 10053 | 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 10054 | Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 10106. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝐴 ∈ GCH ∧ (𝐴 ⊔ 𝐴) ≈ 𝐴 ∧ 𝐵 ≼ 𝒫 𝐴) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)) | ||
Theorem | fpwwe2cbv 10055* | Lemma for fpwwe2 10068. (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 10056* | Lemma for fpwwe2 10068. (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 10057* | Lemma for fpwwe2 10068. (Contributed by Mario Carneiro, 19-May-2015.) |
⊢ 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 10058* | Lemma for fpwwe2 10068. (Contributed by Mario Carneiro, 19-May-2015.) |
⊢ 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 | fpwwe2lem5 10059* | Lemma for fpwwe2 10068. (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))} & ⊢ (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 | fpwwe2lem6 10060* | Lemma for fpwwe2 10068. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ 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 | fpwwe2lem7 10061* | Lemma for fpwwe2 10068. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ 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 | fpwwe2lem8 10062* | Lemma for fpwwe2 10068. 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.) |
⊢ 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 | fpwwe2lem9 10063* | Lemma for fpwwe2 10068. 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.) |
⊢ 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 | fpwwe2lem10 10064* | Lemma for fpwwe2 10068. 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〉 ∣ ((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 | fpwwe2lem11 10065* | Lemma for fpwwe2 10068. (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))} & ⊢ (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 | fpwwe2lem12 10066* | Lemma for fpwwe2 10068. (Contributed by Mario Carneiro, 18-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) |
⊢ 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 | fpwwe2lem13 10067* | Lemma for fpwwe2 10068. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ 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 10068* | 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 9459. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ 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 10069* | Lemma for fpwwe 10071. (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 10070* | Lemma for fpwwe 10071. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ 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 10071* | Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 8673 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 9459. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ 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 10072* | An "effective" form of Cantor's theorem canth 7114. 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 10073* | Lemma for canthnum 10074. (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 10074 | The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8673. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ (𝒫 𝐴 ∩ dom card)) | ||
Theorem | canthwelem 10075* | Lemma for canthwe 10076. (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 10076* | The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 8673. 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 10077 | Lemma for canthp1 10079. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 2o) ≼ 𝒫 𝐴) | ||
Theorem | canthp1lem2 10078* | Lemma for canthp1 10079. (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 10079 | 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 10080 | The exclusion of finite sets from consideration in df-gch 10046 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 10081 | 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 10082 | An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝐴) | ||
Theorem | pwfseqlem1 10083* | Lemma for pwfseq 10089. 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 10084* | Lemma for pwfseq 10089. (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 10085* | Lemma for pwfseq 10089. Using the construction 𝐷 from pwfseqlem1 10083, 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 10086* | Lemma for pwfseqlem4 10087. (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 10087* | Lemma for pwfseq 10089. Derive a final contradiction from the function 𝐹 in pwfseqlem3 10085. Applying fpwwe2 10068 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 10088* |
Lemma for pwfseq 10089. Although in some ways pwfseqlem4 10087 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 9456. 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 9443. 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 9172), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 9447). (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 10089* | 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 10090 | 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 10091 | 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 10092 | 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 10093 | 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 10094 | 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 10095 | 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 10096 | 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 10097 | 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 10098 | If 𝐴 + ≈ 𝒫 𝐴, then 𝐴 is a GCH-set. The much simpler converse to gchhar 10104. (Contributed by Mario Carneiro, 2-Jun-2015.) |
⊢ ((har‘𝐴) ≈ 𝒫 𝐴 → 𝐴 ∈ GCH) | ||
Theorem | alephgch 10099 | 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 10100 | 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) |
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