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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tcwf 9301 | 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 9302 | 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 9303* | 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 9304* | 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 9305* | Theorem scheme version of scottex 9303. 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 9306* | Theorem scheme version of scott0 9304. 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 9307* | Lemma for the Collection Principle cp 9309. (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 9308* | Lemma for the Collection Principle cp 9309. (Contributed by NM, 17-Oct-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅) | ||
Theorem | cp 9309* | 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 9303 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 9310* | A very strong generalization of the Axiom of Replacement (compare zfrep6 7647), derived from the Collection Principle cp 9309. 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 9311* | A variant of the Boundedness Axiom bnd 9310 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 9312* | 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 9313* | 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 9962). 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 9312 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 9314* | 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 9315* |
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 1922 and weth 9906, using scottexs 9305 to establish the existence of 𝐴. For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9314. (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 9316 | Extend class notation to include disjoint union of two classes. |
class (𝐴 ⊔ 𝐵) | ||
Syntax | cinl 9317 | Extend class notation to include left injection of a disjoint union. |
class inl | ||
Syntax | cinr 9318 | Extend class notation to include right injection of a disjoint union. |
class inr | ||
Definition | df-dju 9319 | 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 9320 | Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | ||
Definition | df-inr 9321 | Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.) |
⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | ||
Theorem | djueq12 9322 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷)) | ||
Theorem | djueq1 9323 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) | ||
Theorem | djueq2 9324 | Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵)) | ||
Theorem | nfdju 9325 | Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵) | ||
Theorem | djuex 9326 | The disjoint union of sets is a set. For a shorter proof using djuss 9338 see djuexALT 9340. (Contributed by AV, 28-Jun-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V) | ||
Theorem | djuexb 9327 | 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 9328 | Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) | ||
Theorem | djurcl 9329 | Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵)) | ||
Theorem | djulf1o 9330 | 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 9331 | 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 9332 | 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 9333 | 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 9334 | 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 9335 | 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 9336 | 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 9337* | 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 9338 | A disjoint union is a subclass of a Cartesian product. (Contributed by AV, 25-Jun-2022.) |
⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵)) | ||
Theorem | djuunxp 9339 | 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 9340 | Alternate proof of djuex 9326, 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 9341 | 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 9342 | 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 9343 | 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 9344 | 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 9345 | 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 9346 | 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 9347 | 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 9348 | 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 9349* | 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 9350* | 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 9351* | 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 9352* | Universal property of the disjoint union. This theorem shows that the disjoint union, together with the left and right injections df-inl 9320 and df-inr 9321, is the coproduct in the category of sets, see Wikipedia "Coproduct", https://en.wikipedia.org/wiki/Coproduct 9321 (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 9353 | Extend class definition to include the cardinal size function. |
class card | ||
Syntax | cale 9354 | Extend class definition to include the aleph function. |
class ℵ | ||
Syntax | ccf 9355 | Extend class definition to include the cofinality function. |
class cf | ||
Syntax | wacn 9356 | The axiom of choice for limited-length sequences. |
class AC 𝐴 | ||
Definition | df-card 9357* | 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 9957 for its value and cardval2 9409 for a simpler version of its value. The principal theorem relating cardinality to equinumerosity is carden 9962. 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 9358 | 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 9481, alephsuc 9483, and alephlim 9482. 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 9359* | Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). See cfval 9658 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 9360* | 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 9361* | 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 9362 | 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 9363* | 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 9364 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card) | ||
Theorem | ennum 9365 | Equinumerous sets are equi-numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card)) | ||
Theorem | finnum 9366 | 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 9367 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ (𝐴 ∈ On → 𝐴 ∈ dom card) | ||
Theorem | tskwe 9368* | 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 9369 | 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 9370* | 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 9371 | 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 9372 | 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 9373* | The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 9957, 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 9374 | Any ordinal number is equinumerous to its cardinal number. Unlike cardid 9958, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.) |
⊢ (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴) | ||
Theorem | cardonle 9375 | 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 9376 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
⊢ (card‘∅) = ∅ | ||
Theorem | cardidm 9377 | 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 9378* | 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)) | ||
Theorem | ficardom 9379 | 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) | ||
Theorem | ficardid 9380 | A finite set is equinumerous to its cardinal number. (Contributed by Mario Carneiro, 21-Sep-2013.) |
⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴) | ||
Theorem | cardnn 9381 | 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) | ||
Theorem | cardnueq0 9382 | The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.) |
⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅)) | ||
Theorem | cardne 9383 | 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) → ¬ 𝐴 ≈ 𝐵) | ||
Theorem | carden2a 9384 | If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 9385 are meant to replace carden 9962 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) |
⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵) | ||
Theorem | carden2b 9385 | If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9384 are meant to replace carden 9962 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) |
⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵)) | ||
Theorem | card1 9386* | A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.) |
⊢ ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | cardsn 9387 | A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.) |
⊢ (𝐴 ∈ 𝑉 → (card‘{𝐴}) = 1o) | ||
Theorem | carddomi2 9388 | Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 9965, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵)) | ||
Theorem | sdomsdomcardi 9389 | A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.) |
⊢ (𝐴 ≺ (card‘𝐵) → 𝐴 ≺ 𝐵) | ||
Theorem | cardlim 9390 | 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)) | ||
Theorem | cardsdomelir 9391 | A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9392 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.) |
⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵) | ||
Theorem | cardsdomel 9392 | 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) → (𝐴 ≺ 𝐵 ↔ 𝐴 ∈ (card‘𝐵))) | ||
Theorem | iscard 9393* | Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.) |
⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) | ||
Theorem | iscard2 9394* | 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 (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥))) | ||
Theorem | carddom2 9395 | Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 9965, 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) ↔ 𝐴 ≼ 𝐵)) | ||
Theorem | harcard 9396 | The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ (card‘(har‘𝐴)) = (har‘𝐴) | ||
Theorem | cardprclem 9397* | Lemma for cardprc 9398. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ 𝐴 = {𝑥 ∣ (card‘𝑥) = 𝑥} ⇒ ⊢ ¬ 𝐴 ∈ V | ||
Theorem | cardprc 9398 | The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310. In this proof (which does not use AC), we cannot use Cantor's construction canth3 9972 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 8997 to construct (effectively) (ℵ‘suc 𝐴) from (ℵ‘𝐴), which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.) |
⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V | ||
Theorem | carduni 9399* | The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (card‘∪ 𝐴) = ∪ 𝐴)) | ||
Theorem | cardiun 9400* | The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)) |
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