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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | rankeq0 9901 | 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) = ∅) | ||
| Theorem | rankr1id 9902 | 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‘(𝑅1‘𝐴)) = 𝐴) | ||
| Theorem | rankuni 9903 | 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) | ||
| Theorem | rankr1b 9904 | A relationship between rank and 𝑅1. See rankr1a 9876 for the membership version. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ On → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) | ||
| Theorem | ranksuc 9905 | The rank of a successor. (Contributed by NM, 18-Sep-2006.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘suc 𝐴) = suc (rank‘𝐴) | ||
| Theorem | rankuniss 9906 | 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) | ||
| Theorem | rankval4 9907* | 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) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) | ||
| Theorem | rankbnd 9908* | The rank of a set is bounded by a bound for the successor of its members. (Contributed by NM, 18-Sep-2006.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ 𝐵) | ||
| Theorem | rankbnd2 9909* | 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 → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵)) | ||
| Theorem | rankc1 9910* | A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ (rank‘𝐴) = (rank‘∪ 𝐴)) | ||
| Theorem | rankc2 9911* | A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴)) | ||
| Theorem | rankelun 9912 | 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‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷))) | ||
| Theorem | rankelpr 9913 | 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})) | ||
| Theorem | rankelop 9914 | 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〉)) | ||
| Theorem | rankxpl 9915 | A lower bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))) | ||
| Theorem | rankxpu 9916 | 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‘(𝐴 ∪ 𝐵)) | ||
| Theorem | rankfu 9917 | An upper bound on the rank of a function. (Contributed by Gérard Lang, 5-Aug-2018.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹:𝐴⟶𝐵 → (rank‘𝐹) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))) | ||
| Theorem | rankmapu 9918 | An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (rank‘(𝐴 ↑m 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) | ||
| Theorem | rankxplim 9919 | 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 9922 for the successor case. (Contributed by NM, 19-Sep-2006.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵))) | ||
| Theorem | rankxplim2 9920 | 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‘(𝐴 ∪ 𝐵))) | ||
| Theorem | rankxplim3 9921 | 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))) | ||
| Theorem | rankxpsuc 9922 | 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 9919 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵))) | ||
| Theorem | tcwf 9923 | 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)) | ||
| Theorem | tcrank 9924 | 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))) | ||
| Theorem | scottex 9925* | 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.) |
| ⊢ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V | ||
| Theorem | scott0 9926* | 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 = ∅ ↔ {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅) | ||
| Theorem | scottexs 9927* | Theorem scheme version of scottex 9925. 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 | ||
| Theorem | scott0s 9928* | Theorem scheme version of scott0 9926. 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)))} ≠ ∅) | ||
| Theorem | cplem1 9929* | Lemma for the Collection Principle cp 9931. (Contributed by NM, 17-Oct-2003.) |
| ⊢ 𝐶 = {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} & ⊢ 𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶 ⇒ ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅) | ||
| Theorem | cplem2 9930* | Lemma for the Collection Principle cp 9931. (Contributed by NM, 17-Oct-2003.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅) | ||
| Theorem | cp 9931* | 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 9925 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.) |
| ⊢ ∃𝑤∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → ∃𝑦 ∈ 𝑤 𝜑) | ||
| Theorem | bnd 9932* | A very strong generalization of the Axiom of Replacement (compare zfrep6 7979), derived from the Collection Principle cp 9931. 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.) |
| ⊢ (∀𝑥 ∈ 𝑧 ∃𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑) | ||
| Theorem | bnd2 9933* | A variant of the Boundedness Axiom bnd 9932 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 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑)) | ||
| Theorem | kardex 9934* | 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 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V | ||
| Theorem | karden 9935* | 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 10591). 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 9934 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥 ∣ 𝑥 ≈ 𝐴}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐶 = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ≈ 𝐵 ∧ ∀𝑦(𝑦 ≈ 𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ⇒ ⊢ (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵) | ||
| Theorem | htalem 9936* | 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 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝐴 ≠ ∅) → 𝐵 ∈ 𝐴) | ||
| Theorem | hta 9937* |
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
https://us.metamath.org/downloads/choice.txt
(copy of obsolete link
http://ghilbert.org/choice.txt) and
https://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 Hilbert's 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 1930 and weth 10535, using scottexs 9927 to establish the existence of 𝐴. For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9936. (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 = (℩𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑧) ⇒ ⊢ (𝑅 We 𝐴 → (𝜑 → [𝐵 / 𝑥]𝜑)) | ||
| Syntax | cdju 9938 | Extend class notation to include disjoint union of two classes. |
| class (𝐴 ⊔ 𝐵) | ||
| Syntax | cinl 9939 | Extend class notation to include left injection of a disjoint union. |
| class inl | ||
| Syntax | cinr 9940 | Extend class notation to include right injection of a disjoint union. |
| class inr | ||
| Definition | df-dju 9941 | Disjoint union of two classes. This is a way of creating a set which contains elements corresponding to each element of 𝐴 or 𝐵, tagging each one with whether it came from 𝐴 or 𝐵. (Contributed by Jim Kingdon, 20-Jun-2022.) |
| ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | ||
| Definition | df-inl 9942 | Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
| ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | ||
| Definition | df-inr 9943 | Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
| ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | ||
| Theorem | djueq12 9944 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
| ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷)) | ||
| Theorem | djueq1 9945 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
| ⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) | ||
| Theorem | djueq2 9946 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
| ⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵)) | ||
| Theorem | nfdju 9947 | Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
| ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵) | ||
| Theorem | djuex 9948 | The disjoint union of sets is a set. For a shorter proof using djuss 9960 see djuexALT 9962. (Contributed by AV, 28-Jun-2022.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) | ||
| Theorem | djuexb 9949 | The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.) |
| ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V) | ||
| Theorem | djulcl 9950 | Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
| ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) | ||
| Theorem | djurcl 9951 | Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
| ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵)) | ||
| Theorem | djulf1o 9952 | The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
| ⊢ inl:V–1-1-onto→({∅} × V) | ||
| Theorem | djurf1o 9953 | The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.) |
| ⊢ inr:V–1-1-onto→({1o} × V) | ||
| Theorem | inlresf 9954 | The left injection restricted to the left class of a disjoint union is a function from the left class into the disjoint union. (Contributed by AV, 27-Jun-2022.) |
| ⊢ (inl ↾ 𝐴):𝐴⟶(𝐴 ⊔ 𝐵) | ||
| Theorem | inlresf1 9955 | The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.) |
| ⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵) | ||
| Theorem | inrresf 9956 | The right injection restricted to the right class of a disjoint union is a function from the right class into the disjoint union. (Contributed by AV, 27-Jun-2022.) |
| ⊢ (inr ↾ 𝐵):𝐵⟶(𝐴 ⊔ 𝐵) | ||
| Theorem | inrresf1 9957 | The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.) |
| ⊢ (inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵) | ||
| Theorem | djuin 9958 | The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) |
| ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅ | ||
| Theorem | djur 9959* | A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) |
| ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))) | ||
| Theorem | djuss 9960 | A disjoint union is a subclass of a Cartesian product. (Contributed by AV, 25-Jun-2022.) |
| ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) | ||
| Theorem | djuunxp 9961 | The union of a disjoint union and its inversion is the Cartesian product of an unordered pair and the union of the left and right classes of the disjoint unions. (Proposed by GL, 4-Jul-2022.) (Contributed by AV, 4-Jul-2022.) |
| ⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) = ({∅, 1o} × (𝐴 ∪ 𝐵)) | ||
| Theorem | djuexALT 9962 | Alternate proof of djuex 9948, which is shorter, but based indirectly on the definitions of inl and inr. (Proposed by BJ, 28-Jun-2022.) (Contributed by AV, 28-Jun-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) | ||
| Theorem | eldju1st 9963 | The first component of an element of a disjoint union is either ∅ or 1o. (Contributed by AV, 26-Jun-2022.) |
| ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o)) | ||
| Theorem | eldju2ndl 9964 | The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.) |
| ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) = ∅) → (2nd ‘𝑋) ∈ 𝐴) | ||
| Theorem | eldju2ndr 9965 | The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.) |
| ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵) | ||
| Theorem | djuun 9966 | The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) |
| ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵) | ||
| Theorem | 1stinl 9967 | The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.) |
| ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅) | ||
| Theorem | 2ndinl 9968 | The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.) |
| ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋) | ||
| Theorem | 1stinr 9969 | The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.) |
| ⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o) | ||
| Theorem | 2ndinr 9970 | The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.) |
| ⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋) | ||
| Theorem | updjudhf 9971* | The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) & ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶) | ||
| Theorem | updjudhcoinlf 9972* | The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) & ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ⇒ ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹) | ||
| Theorem | updjudhcoinrg 9973* | The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) & ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) ⇒ ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺) | ||
| Theorem | updjud 9974* | Universal property of the disjoint union. This theorem shows that the disjoint union, together with the left and right injections df-inl 9942 and df-inr 9943, is the coproduct in the category of sets, see Wikipedia "Coproduct", https://en.wikipedia.org/wiki/Coproduct 9943 (25-Aug-2023). This is a special case of Example 1 of coproducts in Section 10.67 of [Adamek] p. 185. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ∃!ℎ(ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺)) | ||
| Syntax | ccrd 9975 | Extend class definition to include the cardinal size function. |
| class card | ||
| Syntax | cale 9976 | Extend class definition to include the aleph function. |
| class ℵ | ||
| Syntax | ccf 9977 | Extend class definition to include the cofinality function. |
| class cf | ||
| Syntax | wacn 9978 | The axiom of choice for limited-length sequences. |
| class AC 𝐴 | ||
| Definition | df-card 9979* | 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 10586 for its value and cardval2 10031 for a simpler version of its value. The principal theorem relating cardinality to equinumerosity is carden 10591. 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 ∣ 𝑦 ≈ 𝑥}) | ||
| Definition | df-aleph 9980 | 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 10106, alephsuc 10108, and alephlim 10107. 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) | ||
| Definition | df-cf 9981* | Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 10287 for its value and a description. (Contributed by NM, 1-Apr-2004.) |
| ⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑧 𝑣 ⊆ 𝑢))}) | ||
| Definition | df-acn 9982* | Define a local and length-limited version of the axiom of choice. The definition of the predicate 𝑋 ∈ AC 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)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} | ||
| Theorem | cardf2 9983* | 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 𝑦 ≈ 𝑥}⟶On | ||
| Theorem | cardon 9984 | 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 | ||
| Theorem | isnum2 9985* | 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 𝑥 ≈ 𝐴) | ||
| Theorem | isnumi 9986 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) | ||
| Theorem | ennum 9987 | Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
| ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card)) | ||
| Theorem | finnum 9988 | Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card) | ||
| Theorem | onenon 9989 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
| ⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | ||
| Theorem | tskwe 9990* | A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card) | ||
| Theorem | xpnum 9991 | 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) | ||
| Theorem | cardval3 9992* | 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 ∣ 𝑥 ≈ 𝐴}) | ||
| Theorem | cardid2 9993 | 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) | ||
| Theorem | isnum3 9994 | A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.) |
| ⊢ (𝐴 ∈ dom card ↔ (card‘𝐴) ≈ 𝐴) | ||
| Theorem | oncardval 9995* | The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 10586, 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 ∣ 𝑥 ≈ 𝐴}) | ||
| Theorem | oncardid 9996 | Any ordinal number is equinumerous to its cardinal number. Unlike cardid 10587, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.) |
| ⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴) | ||
| Theorem | cardonle 9997 | 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) | ||
| Theorem | card0 9998 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
| ⊢ (card‘∅) = ∅ | ||
| Theorem | cardidm 9999 | The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.) |
| ⊢ (card‘(card‘𝐴)) = (card‘𝐴) | ||
| Theorem | oncard 10000* | 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)) | ||
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