Home Metamath Proof ExplorerTheorem List (p. 88 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27759) Hilbert Space Explorer (27760-29284) Users' Mathboxes (29285-42322)

Theorem List for Metamath Proof Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembndrank 8701* Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)
(∃𝑥 ∈ On ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)

Theoremunbndrank 8702* The elements of a proper class have unbounded rank. Exercise 2 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.)
𝐴 ∈ V → ∀𝑥 ∈ On ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))

Theoremrankpw 8703 The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 22-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V       (rank‘𝒫 𝐴) = suc (rank‘𝐴)

Theoremranklim 8704 The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.)
(Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵))

Theoremr1pw 8705 A stronger property of 𝑅1 than rankpw 8703. The latter merely proves that 𝑅1 of the successor is a power set, but here we prove that if 𝐴 is in the cumulative hierarchy, then 𝒫 𝐴 is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))

Theoremr1pwALT 8706 Alternate shorter proof of r1pw 8705 based on the additional axioms ax-reg 8494 and ax-inf2 8535. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))

Theoremr1pwcl 8707 The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)
(Lim 𝐵 → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵)))

Theoremrankssb 8708 The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐵 (𝑅1 “ On) → (𝐴𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)))

Theoremrankss 8709 The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐵 ∈ V       (𝐴𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵))

Theoremrankunb 8710 The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Theoremrankprb 8711 The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Theoremrankopb 8712 The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.)
((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)))

Theoremrankuni2b 8713* The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 8-Jun-2013.)
(𝐴 (𝑅1 “ On) → (rank‘ 𝐴) = 𝑥𝐴 (rank‘𝑥))

Theoremranksn 8714 The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V       (rank‘{𝐴}) = suc (rank‘𝐴)

Theoremrankuni2 8715* The rank of a union. Part of Theorem 15.17(iv) of [Monk1] p. 112. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V       (rank‘ 𝐴) = 𝑥𝐴 (rank‘𝑥)

Theoremrankun 8716 The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by NM, 26-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))

Theoremrankpr 8717 The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))

Theoremrankop 8718 The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 13-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (rank‘⟨𝐴, 𝐵⟩) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))

Theoremr1rankid 8719 Any set is a subset of the hierarchy of its rank. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴𝑉𝐴 ⊆ (𝑅1‘(rank‘𝐴)))

Theoremrankeq0b 8720 A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐴 (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅))

Theoremrankeq0 8721 A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V       (𝐴 = ∅ ↔ (rank‘𝐴) = ∅)

Theoremrankr1id 8722 The rank of the hierarchy of an ordinal number is itself. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝐴)) = 𝐴)

Theoremrankuni 8723 The rank of a union. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
(rank‘ 𝐴) = (rank‘𝐴)

Theoremrankr1b 8724 A relationship between rank and 𝑅1. See rankr1a 8696 for the membership version. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V       (𝐵 ∈ On → (𝐴 ⊆ (𝑅1𝐵) ↔ (rank‘𝐴) ⊆ 𝐵))

Theoremranksuc 8725 The rank of a successor. (Contributed by NM, 18-Sep-2006.)
𝐴 ∈ V       (rank‘suc 𝐴) = suc (rank‘𝐴)

Theoremrankuniss 8726 Upper bound of the rank of a union. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 30-Nov-2003.)
𝐴 ∈ V       (rank‘ 𝐴) ⊆ (rank‘𝐴)

Theoremrankval4 8727* The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. (Contributed by NM, 12-Oct-2003.)
𝐴 ∈ V       (rank‘𝐴) = 𝑥𝐴 suc (rank‘𝑥)

Theoremrankbnd 8728* The rank of a set is bounded by a bound for the successor of its members. (Contributed by NM, 18-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 suc (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ 𝐵)

Theoremrankbnd2 8729* The rank of a set is bounded by the successor of a bound for its members. (Contributed by NM, 15-Sep-2006.)
𝐴 ∈ V       (𝐵 ∈ On → (∀𝑥𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))

Theoremrankc1 8730* A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) = (rank‘ 𝐴))

Theoremrankc2 8731* A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)
𝐴 ∈ V       (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) = suc (rank‘ 𝐴))

Theoremrankelun 8732 Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴𝐵)) ∈ (rank‘(𝐶𝐷)))

Theoremrankelpr 8733 Rank membership is inherited by unordered pairs. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷}))

Theoremrankelop 8734 Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘⟨𝐴, 𝐵⟩) ∈ (rank‘⟨𝐶, 𝐷⟩))

Theoremrankxpl 8735 A lower bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))

Theoremrankxpu 8736 An upper bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴𝐵))

Theoremrankfu 8737 An upper bound on the rank of a function. (Contributed by Gérard Lang, 5-Aug-2018.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹:𝐴𝐵 → (rank‘𝐹) ⊆ suc suc (rank‘(𝐴𝐵)))

Theoremrankmapu 8738 An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.)
𝐴 ∈ V    &   𝐵 ∈ V       (rank‘(𝐴𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴𝐵))

Theoremrankxplim 8739 The rank of a Cartesian product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 8742 for the successor case. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))

Theoremrankxplim2 8740 If the rank of a Cartesian product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴𝐵)))

Theoremrankxplim3 8741 The rank of a Cartesian product is a limit ordinal iff its union is. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 × 𝐵)))

Theoremrankxpsuc 8742 The rank of a Cartesian product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 8739 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴𝐵)))

Theoremtcwf 8743 The transitive closure function is well-founded if its argument is. (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴 (𝑅1 “ On) → (TC‘𝐴) ∈ (𝑅1 “ On))

Theoremtcrank 8744 This theorem expresses two different facts from the two subset implications in this equality. In the forward direction, it says that the transitive closure has members of every rank below 𝐴. Stated another way, to construct a set at a given rank, you have to climb the entire hierarchy of ordinals below (rank‘𝐴), constructing at least one set at each level in order to move up the ranks. In the reverse direction, it says that every member of (TC‘𝐴) has a rank below the rank of 𝐴, since intuitively it contains only the members of 𝐴 and the members of those and so on, but nothing "bigger" than 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴 (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))

2.6.6  Scott's trick; collection principle; Hilbert's epsilon

Theoremscottex 8745* Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set. (Contributed by NM, 13-Oct-2003.)
{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V

Theoremscott0 8746* Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. 𝐴 is empty). (Contributed by NM, 15-Oct-2003.)
(𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)

Theoremscottexs 8747* Theorem scheme version of scottex 8745. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is a set. (Contributed by NM, 13-Oct-2003.)
{𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V

Theoremscott0s 8748* Theorem scheme version of scott0 8746. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is not empty iff there is an 𝑥 such that 𝜑(𝑥) holds. (Contributed by NM, 13-Oct-2003.)
(∃𝑥𝜑 ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)

Theoremcplem1 8749* Lemma for the Collection Principle cp 8751. (Contributed by NM, 17-Oct-2003.)
𝐶 = {𝑦𝐵 ∣ ∀𝑧𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)}    &   𝐷 = 𝑥𝐴 𝐶       𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝐷) ≠ ∅)

Theoremcplem2 8750* -Lemma for the Collection Principle cp 8751. (Contributed by NM, 17-Oct-2003.)
𝐴 ∈ V       𝑦𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅)

Theoremcp 8751* Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 8745 that collapses a proper class into a set of minimum rank. The wff 𝜑 can be thought of as 𝜑(𝑥, 𝑦). Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.)
𝑤𝑥𝑧 (∃𝑦𝜑 → ∃𝑦𝑤 𝜑)

Theorembnd 8752* A very strong generalization of the Axiom of Replacement (compare zfrep6 7131), derived from the Collection Principle cp 8751. Its strength lies in the rather profound fact that 𝜑(𝑥, 𝑦) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. (Contributed by NM, 17-Oct-2004.)
(∀𝑥𝑧𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)

Theorembnd2 8753* A variant of the Boundedness Axiom bnd 8752 that picks a subset 𝑧 out of a possibly proper class 𝐵 in which a property is true. (Contributed by NM, 4-Feb-2004.)
𝐴 ∈ V       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑))

Theoremkardex 8754* The collection of all sets equinumerous to a set 𝐴 and having the least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.)
{𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V

Theoremkarden 8755* If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 9370). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 8754 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥𝑥𝐴}. (Contributed by NM, 18-Dec-2003.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}    &   𝐷 = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}       (𝐶 = 𝐷𝐴𝐵)

Theoremhtalem 8756* Lemma for defining an emulation of Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem is equivalent to Hilbert's "transfinite axiom," described on that page, with the additional 𝑅 We 𝐴 antecedent. The element 𝐵 is the epsilon that the theorem emulates. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)
𝐴 ∈ V    &   𝐵 = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)       ((𝑅 We 𝐴𝐴 ≠ ∅) → 𝐵𝐴)

Theoremhta 8757* A ZFC emulation of Hilbert's transfinite axiom. The set 𝐵 has the properties of Hilbert's epsilon, except that it also depends on a well-ordering 𝑅. This theorem arose from discussions with Raph Levien on 5-Mar-2004 about translating the HOL proof language, which uses Hilbert's epsilon. See http://us.metamath.org/downloads/choice.txt (copy of obsolete link http://ghilbert.org/choice.txt) and http://us.metamath.org/downloads/megillaward2005he.pdf.

Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires 𝑅 We 𝐴 as an antecedent. Class 𝐴 collects the sets of the least rank for which 𝜑(𝑥) is true. Class 𝐵, which emulates the epsilon, is the minimum element in a well-ordering 𝑅 on 𝐴.

If a well-ordering 𝑅 on 𝐴 can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace 𝑅 with a dummy setvar variable, say 𝑤, and attach 𝑤 We 𝐴 as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, 𝐵 (which will have 𝑤 as a free variable) will no longer be present, and we can eliminate 𝑤 We 𝐴 by applying exlimiv 1857 and weth 9314, using scottexs 8747 to establish the existence of 𝐴.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 8756. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.)

𝐴 = {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))}    &   𝐵 = (𝑧𝐴𝑤𝐴 ¬ 𝑤𝑅𝑧)       (𝑅 We 𝐴 → (𝜑[𝐵 / 𝑥]𝜑))

2.6.7  Cardinal numbers

Syntaxccrd 8758 Extend class definition to include the cardinal size function.
class card

Syntaxcale 8759 Extend class definition to include the aleph function.
class

Syntaxccf 8760 Extend class definition to include the cofinality function.
class cf

Syntaxwacn 8761 The axiom of choice for limited-length sequences.
class AC 𝐴

Definitiondf-card 8762* Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. See cardval 9365 for its value, cardval2 8814 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 9370. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
card = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})

Definitiondf-aleph 8763 Define the aleph function. Our definition expresses Definition 12 of [Suppes] p. 229 in a closed form, from which we derive the recursive definition as theorems aleph0 8886, alephsuc 8888, and alephlim 8887. The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it. (Contributed by NM, 21-Oct-2003.)
ℵ = rec(har, ω)

Definitiondf-cf 8764* Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 9066 for its value and a description. (Contributed by NM, 1-Apr-2004.)
cf = (𝑥 ∈ On ↦ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑣𝑥𝑢𝑧 𝑣𝑢))})

Definitiondf-acn 8765* Define a local and length-limited version of the axiom of choice. The definition of the predicate 𝑋AC 𝐴 is that for all families of nonempty subsets of 𝑋 indexed on 𝐴 (i.e. functions 𝐴⟶𝒫 𝑋 ∖ {∅}), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.)
AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 𝐴)∃𝑔𝑦𝐴 (𝑔𝑦) ∈ (𝑓𝑦))}

Theoremcardf2 8766* The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 20-Sep-2014.)
card:{𝑥 ∣ ∃𝑦 ∈ On 𝑦𝑥}⟶On

Theoremcardon 8767 The cardinal number of a set is an ordinal number. Proposition 10.6(1) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
(card‘𝐴) ∈ On

Theoremisnum2 8768* A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.)
(𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)

Theoremisnumi 8769 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ On ∧ 𝐴𝐵) → 𝐵 ∈ dom card)

Theoremennum 8770 Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
(𝐴𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))

Theoremfinnum 8771 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ Fin → 𝐴 ∈ dom card)

Theoremonenon 8772 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ On → 𝐴 ∈ dom card)

Theoremtskwe 8773* A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉 ∧ {𝑥 ∈ 𝒫 𝐴𝑥𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)

Theoremxpnum 8774 The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)

Theoremcardval3 8775* An alternate definition of the value of (card‘𝐴) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
(𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})

Theoremcardid2 8776 Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
(𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)

Theoremisnum3 8777 A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ dom card ↔ (card‘𝐴) ≈ 𝐴)

Theoremoncardval 8778* The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 9365, this theorem does not require the Axiom of Choice. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
(𝐴 ∈ On → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})

Theoremoncardid 8779 Any ordinal number is equinumerous to its cardinal number. Unlike cardid 9366, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)
(𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)

Theoremcardonle 8780 The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
(𝐴 ∈ On → (card‘𝐴) ⊆ 𝐴)

Theoremcard0 8781 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
(card‘∅) = ∅

Theoremcardidm 8782 The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
(card‘(card‘𝐴)) = (card‘𝐴)

Theoremoncard 8783* A set is a cardinal number iff it equals its own cardinal number. Proposition 10.9 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
(∃𝑥 𝐴 = (card‘𝑥) ↔ 𝐴 = (card‘𝐴))

Theoremficardom 8784 The cardinal number of a finite set is a finite ordinal. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 4-Feb-2013.)
(𝐴 ∈ Fin → (card‘𝐴) ∈ ω)

Theoremficardid 8785 A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.)
(𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)

Theoremcardnn 8786 The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90. (Contributed by Mario Carneiro, 7-Jan-2013.)
(𝐴 ∈ ω → (card‘𝐴) = 𝐴)

Theoremcardnueq0 8787 The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.)
(𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))

Theoremcardne 8788 No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
(𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)

Theoremcarden2a 8789 If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 8790 are meant to replace carden 9370 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)
(((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴𝐵)

Theoremcarden2b 8790 If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 8789 are meant to replace carden 9370 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
(𝐴𝐵 → (card‘𝐴) = (card‘𝐵))

Theoremcard1 8791* A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)
((card‘𝐴) = 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})

Theoremcardsn 8792 A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)
(𝐴𝑉 → (card‘{𝐴}) = 1𝑜)

Theoremcarddomi2 8793 Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 9373, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))

Theoremsdomsdomcardi 8794 A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.)
(𝐴 ≺ (card‘𝐵) → 𝐴𝐵)

Theoremcardlim 8795 An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.)
(ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴))

Theoremcardsdomelir 8796 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 8797 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)
(𝐴 ∈ (card‘𝐵) → 𝐴𝐵)

Theoremcardsdomel 8797 A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴𝐵𝐴 ∈ (card‘𝐵)))

Theoremiscard 8798* Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)
((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 𝑥𝐴))

Theoremiscard2 8799* Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.)
((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))

Theoremcarddom2 8800 Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 9373, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
 Copyright terms: Public domain < Previous  Next >