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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | canth3 10601 | Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.) |
| ⊢ (𝐴 ∈ 𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴)) | ||
| Theorem | infxpidm 10602 | Every infinite class is equinumerous to its Cartesian square. This theorem, which is equivalent to the axiom of choice over ZF, provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This is a corollary of infxpen 10054 (used via infxpidm2 10057). (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.) |
| ⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴) | ||
| Theorem | ondomon 10603* | The class of ordinals dominated by a given set is an ordinal. Theorem 56 of [Suppes] p. 227. This theorem can be proved without the axiom of choice, see hartogs 9584. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.) Use hartogs 9584 instead. (New usage is discouraged.) |
| ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On) | ||
| Theorem | cardmin 10604* | The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.) |
| ⊢ (𝐴 ∈ 𝑉 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) | ||
| Theorem | ficard 10605 | A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω)) | ||
| Theorem | infinf 10606 | Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.) |
| ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴)) | ||
| Theorem | unirnfdomd 10607 | The union of the range of a function from an infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.) |
| ⊢ (𝜑 → 𝐹:𝑇⟶Fin) & ⊢ (𝜑 → ¬ 𝑇 ∈ Fin) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇) | ||
| Theorem | konigthlem 10608* | Lemma for konigth 10609. (Contributed by Mario Carneiro, 22-Feb-2013.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖) & ⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖) & ⊢ 𝐷 = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))) & ⊢ 𝐸 = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖)) ⇒ ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃) | ||
| Theorem | konigth 10609* | Konig's Theorem. If 𝑚(𝑖) ≺ 𝑛(𝑖) for all 𝑖 ∈ 𝐴, then Σ𝑖 ∈ 𝐴𝑚(𝑖) ≺ ∏𝑖 ∈ 𝐴𝑛(𝑖), where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with unions rather than disjoint unions for convenience, but the version with disjoint unions is clearly a special case of this version. The Axiom of Choice is needed for this proof, but it contains AC as a simple corollary (letting 𝑚(𝑖) = ∅, this theorem says that an infinite cartesian product of nonempty sets is nonempty), so this is an AC equivalent. Theorem 11.26 of [TakeutiZaring] p. 107. (Contributed by Mario Carneiro, 22-Feb-2013.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖) & ⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖) ⇒ ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃) | ||
| Theorem | alephsucpw 10610 | The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 10716 or gchaleph2 10712.) (Contributed by NM, 27-Aug-2005.) |
| ⊢ (ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) | ||
| Theorem | aleph1 10611 | The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.) (Contributed by NM, 7-Jul-2004.) |
| ⊢ (ℵ‘1o) ≼ (2o ↑m (ℵ‘∅)) | ||
| Theorem | alephval2 10612* | An alternate way to express the value of the aleph function for nonzero arguments. Theorem 64 of [Suppes] p. 229. (Contributed by NM, 15-Nov-2003.) |
| ⊢ ((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥}) | ||
| Theorem | dominfac 10613 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 10499. See dominf 10485 for a version proved from ax-cc 10475. (Contributed by NM, 25-Mar-2007.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴) | ||
| Theorem | iunctb 10614* | The countable union of countable sets is countable (indexed union version of unictb 10615). (Contributed by Mario Carneiro, 18-Jan-2014.) |
| ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ ω) | ||
| Theorem | unictb 10615* | The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 10614 for indexed union version. (Contributed by NM, 26-Mar-2006.) |
| ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ω) → ∪ 𝐴 ≼ ω) | ||
| Theorem | infmap 10616* | 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 10617 | 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 10618 | 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 10619 | 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 10620* | An alternate representation of a successor aleph. Compare alephsuc 10108 and alephsuc2 10120. Equality can be obtained by taking the card of the right-hand side then using alephcard 10110 and carden 10591. (Contributed by NM, 23-Oct-2004.) |
| ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}) | ||
| Theorem | alephexp2 10621* | An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 10619 (which works if the base is less than or equal to the exponent) and infmap 10616 (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 10622 | 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 10623* | A corollary of Konig's Theorem konigth 10609. 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 10624 | A corollary of Konig's Theorem konigth 10609. 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 10625 | From canth2 9170, 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 10609 (in the form of cfpwsdom 10624), (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 10626 | 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 10627 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) |
| ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧) | ||
| Theorem | nd2 10628 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) |
| ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) | ||
| Theorem | nd3 10629 | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.) |
| ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦) | ||
| Theorem | nd4 10630 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
| ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥) | ||
| Theorem | axextnd 10631 | A version of the Axiom of Extensionality with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 14-Aug-2003.) (New usage is discouraged.) |
| ⊢ ∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) | ||
| Theorem | axrepndlem1 10632* | Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10633 | Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10634 | A version of the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10635* | Lemma for the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
| ⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
| Theorem | axunnd 10636 | A version of the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
| ⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
| Theorem | axpowndlem1 10637 | 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 10638* | 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 2377. (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 10639* | Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10640 | Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10641 | 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 2377. (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 10642 | Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10643* | Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10644 | A version of the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10645* | 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 10646 | 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 10647 | Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10648 | Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10649 | Lemma for the Axiom of Choice with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10650* | 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 10651* | 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 10652 | 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 10653* | Axiom of Extensionality ax-ext 2708, reproved from conditionless ZFC version and predicate calculus. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | ||
| Theorem | zfcndrep 10654* | Axiom of Replacement ax-rep 5279, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10655* | Axiom of Union ax-un 7755, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10656* | Axiom of Power Sets ax-pow 5365, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness" dtru 5441. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10657* | Axiom of Regularity ax-reg 9632, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 10658* | Axiom of Infinity ax-inf 9678, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing Theorem el 5442 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 10659* | Axiom of Choice ax-ac 10499, 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 10660 | Extend class notation to include the collection of sets that satisfy the GCH. |
| class GCH | ||
| Definition | df-gch 10661* | 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 10662* | 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 10663 | A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.) |
| ⊢ Fin ⊆ GCH | ||
| Theorem | gchi 10664 | 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 10665 | 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 10666 | 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 10667 | 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 10668 | 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 10669 | Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 10721. (Contributed by Mario Carneiro, 15-May-2015.) |
| ⊢ ((𝐴 ∈ GCH ∧ (𝐴 ⊔ 𝐴) ≈ 𝐴 ∧ 𝐵 ≼ 𝒫 𝐴) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)) | ||
| Theorem | fpwwe2cbv 10670* | Lemma for fpwwe2 10683. (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 10671* | Lemma for fpwwe2 10683. (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 10672* | Lemma for fpwwe2 10683. (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 10673* | Lemma for fpwwe2 10683. (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 10674* | Lemma for fpwwe2 10683. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.) (Proof shortened by Matthew House, 10-Sep-2025.) |
| ⊢ 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 10675* | Lemma for fpwwe2 10683. (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 10676* | Lemma for fpwwe2 10683. (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 10677* | Lemma for fpwwe2 10683. 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 10678* | Lemma for fpwwe2 10683. 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 10679* | Lemma for fpwwe2 10683. 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 10680* | Lemma for fpwwe2 10683. (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 10681* | Lemma for fpwwe2 10683. (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 10682* | Lemma for fpwwe2 10683. (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 10683* | 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 10070. 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 10684* | Lemma for fpwwe 10686. (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 10685* | Lemma for fpwwe 10686. (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 10686* | Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 9170 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 10070. 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 10687* | An "effective" form of Cantor's theorem canth 7385. 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 10688* | Lemma for canthnum 10689. (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 10689 | The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9170. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.) |
| ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ (𝒫 𝐴 ∩ dom card)) | ||
| Theorem | canthwelem 10690* | Lemma for canthwe 10691. (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 10691* | The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9170. 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 10692 | Lemma for canthp1 10694. (Contributed by Mario Carneiro, 18-May-2015.) |
| ⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 2o) ≼ 𝒫 𝐴) | ||
| Theorem | canthp1lem2 10693* | Lemma for canthp1 10694. (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 10694 | 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 10695 | The exclusion of finite sets from consideration in df-gch 10661 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 10696 | 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 10697 | An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.) |
| ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝐴) | ||
| Theorem | pwfseqlem1 10698* | Lemma for pwfseq 10704. 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 10699* | Lemma for pwfseq 10704. (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 10700* | Lemma for pwfseq 10704. Using the construction 𝐷 from pwfseqlem1 10698, 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)) | ||
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