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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | r1limwun 10501 | Each limit stage in the cumulative hierarchy is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (𝑅1‘𝐴) ∈ WUni) | ||
| Theorem | r1wunlim 10502 | The weak universes in the cumulative hierarchy are exactly the limit ordinals. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ (𝐴 ∈ 𝑉 → ((𝑅1‘𝐴) ∈ WUni ↔ Lim 𝐴)) | ||
| Theorem | wunex2 10503* | Construct a weak universe from a given set. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) & ⊢ 𝑈 = ∪ ran 𝐹 ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈)) | ||
| Theorem | wunex 10504* | Construct a weak universe from a given set. See also wunex2 10503. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ (𝐴 ∈ 𝑉 → ∃𝑢 ∈ WUni 𝐴 ⊆ 𝑢) | ||
| Theorem | uniwun 10505 | Every set is contained in a weak universe. This is the analogue of grothtsk 10600 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 10600. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ ∪ WUni = V | ||
| Theorem | wunex3 10506 | Construct a weak universe from a given set. This version of wunex 10504 has a simpler proof, but requires the axiom of regularity. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝑈 = (𝑅1‘((rank‘𝐴) +o ω)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈)) | ||
| Theorem | wuncval 10507* | Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢}) | ||
| Theorem | wuncid 10508 | The weak universe closure of a set contains the set. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (wUniCl‘𝐴)) | ||
| Theorem | wunccl 10509 | The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) ∈ WUni) | ||
| Theorem | wuncss 10510 | The weak universe closure is a subset of any other weak universe containing the set. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ ((𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈) → (wUniCl‘𝐴) ⊆ 𝑈) | ||
| Theorem | wuncidm 10511 | The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴)) | ||
| Theorem | wuncval2 10512* | Our earlier expression for a containing weak universe is in fact the weak universe closure. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω) & ⊢ 𝑈 = ∪ ran 𝐹 ⇒ ⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = 𝑈) | ||
| Syntax | ctsk 10513 | Extend class definition to include the class of all Tarski classes. |
| class Tarski | ||
| Definition | df-tsk 10514* | The class of all Tarski classes. Tarski classes is a phrase coined by Grzegorz Bancerek in his article Tarski's Classes and Ranks, Journal of Formalized Mathematics, Vol 1, No 3, May-August 1990. A Tarski class is a set whose existence is ensured by Tarski's axiom A (see ax-groth 10588 and the equivalent axioms). Axiom A was first presented in Tarski's article Ueber unerreichbare Kardinalzahlen. Tarski introduced the axiom A to enable ZFC to manage inaccessible cardinals. Later Grothendieck introduced the concept of Grothendieck universes and showed they were equal to transitive Tarski classes. (Contributed by FL, 30-Dec-2010.) |
| ⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))} | ||
| Theorem | eltskg 10515* | Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.) |
| ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ ∃𝑤 ∈ 𝑇 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)))) | ||
| Theorem | eltsk2g 10516* | Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.) |
| ⊢ (𝑇 ∈ 𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧 ∈ 𝑇 (𝒫 𝑧 ⊆ 𝑇 ∧ 𝒫 𝑧 ∈ 𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧 ≈ 𝑇 ∨ 𝑧 ∈ 𝑇)))) | ||
| Theorem | tskpwss 10517 | First axiom of a Tarski class. The subsets of an element of a Tarski class belong to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ⊆ 𝑇) | ||
| Theorem | tskpw 10518 | Second axiom of a Tarski class. The powerset of an element of a Tarski class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝒫 𝐴 ∈ 𝑇) | ||
| Theorem | tsken 10519 | Third axiom of a Tarski class. A subset of a Tarski class is either equipotent to the class or an element of the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇) → (𝐴 ≈ 𝑇 ∨ 𝐴 ∈ 𝑇)) | ||
| Theorem | 0tsk 10520 | The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.) |
| ⊢ ∅ ∈ Tarski | ||
| Theorem | tsksdom 10521 | An element of a Tarski class is strictly dominated by the class. JFM CLASSES2 th. 1. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝐴 ≺ 𝑇) | ||
| Theorem | tskssel 10522 | A part of a Tarski class strictly dominated by the class is an element of the class. JFM CLASSES2 th. 2. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇) | ||
| Theorem | tskss 10523 | The subsets of an element of a Tarski class belong to the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑇) | ||
| Theorem | tskin 10524 | The intersection of two elements of a Tarski class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (𝐴 ∩ 𝐵) ∈ 𝑇) | ||
| Theorem | tsksn 10525 | A singleton of an element of a Tarski class belongs to the class. JFM CLASSES2 th. 2 (partly). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 18-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝐴} ∈ 𝑇) | ||
| Theorem | tsktrss 10526 | A transitive element of a Tarski class is a part of the class. JFM CLASSES2 th. 8. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 20-Sep-2014.) |
| ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝐴 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇) | ||
| Theorem | tsksuc 10527 | If an element of a Tarski class is an ordinal number, its successor is an element of the class. JFM CLASSES2 th. 6 (partly). (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ∈ 𝑇) → suc 𝐴 ∈ 𝑇) | ||
| Theorem | tsk0 10528 | A nonempty Tarski class contains the empty set. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∅ ∈ 𝑇) | ||
| Theorem | tsk1 10529 | One is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o ∈ 𝑇) | ||
| Theorem | tsk2 10530 | Two is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2o ∈ 𝑇) | ||
| Theorem | 2domtsk 10531 | If a Tarski class is not empty, it has more than two elements. (Contributed by FL, 22-Feb-2011.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2o ≺ 𝑇) | ||
| Theorem | tskr1om 10532 | A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 9405.) (Contributed by Mario Carneiro, 24-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇) | ||
| Theorem | tskr1om2 10533 | A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 9405.) (Contributed by NM, 22-Feb-2011.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∪ (𝑅1 “ ω) ⊆ 𝑇) | ||
| Theorem | tskinf 10534 | A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇) | ||
| Theorem | tskpr 10535 | If 𝐴 and 𝐵 are members of a Tarski class, their unordered pair is also an element of the class. JFM CLASSES2 th. 3 (partly). (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → {𝐴, 𝐵} ∈ 𝑇) | ||
| Theorem | tskop 10536 | If 𝐴 and 𝐵 are members of a Tarski class, their ordered pair is also an element of the class. JFM CLASSES2 th. 4. (Contributed by FL, 22-Feb-2011.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → ⟨𝐴, 𝐵⟩ ∈ 𝑇) | ||
| Theorem | tskxpss 10537 | A Cartesian product of two parts of a Tarski class is a part of the class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ⊆ 𝑇 ∧ 𝐵 ⊆ 𝑇) → (𝐴 × 𝐵) ⊆ 𝑇) | ||
| Theorem | tskwe2 10538 | A Tarski class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.) |
| ⊢ (𝑇 ∈ Tarski → 𝑇 ∈ dom card) | ||
| Theorem | inttsk 10539 | The intersection of a collection of Tarski classes is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
| ⊢ ((𝐴 ⊆ Tarski ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Tarski) | ||
| Theorem | inar1 10540 | (𝑅1‘𝐴) for 𝐴 a strongly inaccessible cardinal is equipotent to 𝐴. (Contributed by Mario Carneiro, 6-Jun-2013.) |
| ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ≈ 𝐴) | ||
| Theorem | r1omALT 10541 | Alternate proof of r1om 10009, shorter as a consequence of inar1 10540, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝑅1‘ω) ≈ ω | ||
| Theorem | rankcf 10542 | Any set must be at least as large as the cofinality of its rank, because the ranks of the elements of 𝐴 form a cofinal map into (rank‘𝐴). (Contributed by Mario Carneiro, 27-May-2013.) |
| ⊢ ¬ 𝐴 ≺ (cf‘(rank‘𝐴)) | ||
| Theorem | inatsk 10543 | (𝑅1‘𝐴) for 𝐴 a strongly inaccessible cardinal is a Tarski class. (Contributed by Mario Carneiro, 8-Jun-2013.) |
| ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ∈ Tarski) | ||
| Theorem | r1omtsk 10544 | The set of hereditarily finite sets is a Tarski class. (The Tarski-Grothendieck Axiom is not needed for this theorem.) (Contributed by Mario Carneiro, 28-May-2013.) |
| ⊢ (𝑅1‘ω) ∈ Tarski | ||
| Theorem | tskord 10545 | A Tarski class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴 ≺ 𝑇) → 𝐴 ∈ 𝑇) | ||
| Theorem | tskcard 10546 | An even more direct relationship than r1tskina 10547 to get an inaccessible cardinal out of a Tarski class: the size of any nonempty Tarski class is an inaccessible cardinal. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (card‘𝑇) ∈ Inacc) | ||
| Theorem | r1tskina 10547 | There is a direct relationship between transitive Tarski classes and inaccessible cardinals: the Tarski classes that occur in the cumulative hierarchy are exactly at the strongly inaccessible cardinals. (Contributed by Mario Carneiro, 8-Jun-2013.) |
| ⊢ (𝐴 ∈ On → ((𝑅1‘𝐴) ∈ Tarski ↔ (𝐴 = ∅ ∨ 𝐴 ∈ Inacc))) | ||
| Theorem | tskuni 10548 | The union of an element of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 22-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ∈ 𝑇) | ||
| Theorem | tskwun 10549 | A nonempty transitive Tarski class is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝑇 ≠ ∅) → 𝑇 ∈ WUni) | ||
| Theorem | tskint 10550 | The intersection of an element of a transitive Tarski class is an element of the class. (Contributed by FL, 17-Apr-2011.) (Revised by Mario Carneiro, 20-Sep-2014.) |
| ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑇) | ||
| Theorem | tskun 10551 | The union of two elements of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 20-Sep-2014.) |
| ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 ∪ 𝐵) ∈ 𝑇) | ||
| Theorem | tskxp 10552 | The Cartesian product of two elements of a transitive Tarski class is an element of the class. JFM CLASSES2 th. 67 (partly). (Contributed by FL, 15-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
| ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 × 𝐵) ∈ 𝑇) | ||
| Theorem | tskmap 10553 | Set exponentiation is an element of a transitive Tarski class. JFM CLASSES2 th. 67 (partly). (Contributed by FL, 15-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
| ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 ↑m 𝐵) ∈ 𝑇) | ||
| Theorem | tskurn 10554 | A transitive Tarski class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013.) |
| ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹:𝐴⟶𝑇) → ∪ ran 𝐹 ∈ 𝑇) | ||
| Syntax | cgru 10555 | Extend class notation to include the class of all Grothendieck universes. |
| class Univ | ||
| Definition | df-gru 10556* | A Grothendieck universe is a set that is closed with respect to all the operations that are common in set theory: pairs, powersets, unions, intersections, Cartesian products etc. Grothendieck and alii, Séminaire de Géométrie Algébrique 4, Exposé I, p. 185. It was designed to give a precise meaning to the concepts of categories of sets, groups... (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑢))} | ||
| Theorem | elgrug 10557* | Properties of a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)))) | ||
| Theorem | grutr 10558 | A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ (𝑈 ∈ Univ → Tr 𝑈) | ||
| Theorem | gruelss 10559 | A Grothendieck universe is transitive, so each element is a subset of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝐴 ⊆ 𝑈) | ||
| Theorem | grupw 10560 | A Grothendieck universe contains the powerset of each of its members. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → 𝒫 𝐴 ∈ 𝑈) | ||
| Theorem | gruss 10561 | Any subset of an element of a Grothendieck universe is also an element. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝑈) | ||
| Theorem | grupr 10562 | A Grothendieck universe contains pairs derived from its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈) | ||
| Theorem | gruurn 10563 | A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10564 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈) | ||
| Theorem | gruiun 10564* | If 𝐵(𝑥) is a family of elements of 𝑈 and the index set 𝐴 is an element of 𝑈, then the indexed union ∪ 𝑥 ∈ 𝐴𝐵 is also an element of 𝑈, where 𝑈 is a Grothendieck universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) | ||
| Theorem | gruuni 10565 | A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝐴 ∈ 𝑈) | ||
| Theorem | grurn 10566 | A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10564 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈) | ||
| Theorem | gruima 10567 | A Grothendieck universe contains image sets drawn from its members. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ Fun 𝐹 ∧ (𝐹 “ 𝐴) ⊆ 𝑈) → (𝐴 ∈ 𝑈 → (𝐹 “ 𝐴) ∈ 𝑈)) | ||
| Theorem | gruel 10568 | Any element of an element of a Grothendieck universe is also an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝑈) | ||
| Theorem | grusn 10569 | A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈) | ||
| Theorem | gruop 10570 | A Grothendieck universe contains ordered pairs of its elements. (Contributed by Mario Carneiro, 10-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ ∈ 𝑈) | ||
| Theorem | gruun 10571 | A Grothendieck universe contains binary unions of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ∪ 𝐵) ∈ 𝑈) | ||
| Theorem | gruxp 10572 | A Grothendieck universe contains binary cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 × 𝐵) ∈ 𝑈) | ||
| Theorem | grumap 10573 | A Grothendieck universe contains all powers of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → (𝐴 ↑m 𝐵) ∈ 𝑈) | ||
| Theorem | gruixp 10574* | A Grothendieck universe contains indexed cartesian products of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → X𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) | ||
| Theorem | gruiin 10575* | A Grothendieck universe contains indexed intersections of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑈) | ||
| Theorem | gruf 10576 | A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → 𝐹 ∈ 𝑈) | ||
| Theorem | gruen 10577 | A Grothendieck universe contains all subsets of itself that are equipotent to an element of the universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ⊆ 𝑈 ∧ (𝐵 ∈ 𝑈 ∧ 𝐵 ≈ 𝐴)) → 𝐴 ∈ 𝑈) | ||
| Theorem | gruwun 10578 | A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni) | ||
| Theorem | intgru 10579 | The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
| ⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ) | ||
| Theorem | ingru 10580* | The intersection of a universe with a class that acts like a universe is another universe. (Contributed by Mario Carneiro, 10-Jun-2013.) |
| ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝒫 𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥⟶𝐴 → ∪ ran 𝑦 ∈ 𝐴))) → (𝑈 ∈ Univ → (𝑈 ∩ 𝐴) ∈ Univ)) | ||
| Theorem | wfgru 10581 | The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.) |
| ⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ) | ||
| Theorem | grudomon 10582 | Each ordinal that is comparable with an element of the universe is in the universe. (Contributed by Mario Carneiro, 10-Jun-2013.) |
| ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝐵 ∈ 𝑈 ∧ 𝐴 ≼ 𝐵)) → 𝐴 ∈ 𝑈) | ||
| Theorem | gruina 10583 | If a Grothendieck universe 𝑈 is nonempty, then the height of the ordinals in 𝑈 is a strongly inaccessible cardinal. (Contributed by Mario Carneiro, 17-Jun-2013.) |
| ⊢ 𝐴 = (𝑈 ∩ On) ⇒ ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc) | ||
| Theorem | grur1a 10584 | A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.) |
| ⊢ 𝐴 = (𝑈 ∩ On) ⇒ ⊢ (𝑈 ∈ Univ → (𝑅1‘𝐴) ⊆ 𝑈) | ||
| Theorem | grur1 10585 | A characterization of Grothendieck universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.) |
| ⊢ 𝐴 = (𝑈 ∩ On) ⇒ ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → 𝑈 = (𝑅1‘𝐴)) | ||
| Theorem | grutsk1 10586 | Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 10548.) (Contributed by Mario Carneiro, 17-Jun-2013.) |
| ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ) | ||
| Theorem | grutsk 10587 | Grothendieck universes are the same as transitive Tarski classes. (The proof in the forward direction requires Foundation.) (Contributed by Mario Carneiro, 24-Jun-2013.) |
| ⊢ Univ = {𝑥 ∈ Tarski ∣ Tr 𝑥} | ||
| Axiom | ax-groth 10588* | The Tarski-Grothendieck Axiom. For every set 𝑥 there is an inaccessible cardinal 𝑦 such that 𝑦 is not in 𝑥. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics". This version of the axiom is used by the Mizar project (http://www.mizar.org/JFM/Axiomatics/tarski.html). Unlike the ZFC axioms, this axiom is very long when expressed in terms of primitive symbols (see grothprim 10599). An open problem is finding a shorter equivalent. (Contributed by NM, 18-Mar-2007.) |
| ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))) | ||
| Theorem | axgroth5 10589* | The Tarski-Grothendieck axiom using abbreviations. (Contributed by NM, 22-Jun-2009.) |
| ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦)) | ||
| Theorem | axgroth2 10590* | Alternate version of the Tarski-Grothendieck Axiom. (Contributed by NM, 18-Mar-2007.) |
| ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑦 ≼ 𝑧 ∨ 𝑧 ∈ 𝑦))) | ||
| Theorem | grothpw 10591* | Derive the Axiom of Power Sets ax-pow 5289 from the Tarski-Grothendieck axiom ax-groth 10588. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 10588. Note that ax-pow 5289 is not used by the proof. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.) |
| ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | ||
| Theorem | grothpwex 10592 | Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10588. Note that ax-pow 5289 is not used by the proof. Use axpweq 5288 to obtain ax-pow 5289. Use pwex 5304 or pwexg 5302 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.) |
| ⊢ 𝒫 𝑥 ∈ V | ||
| Theorem | axgroth6 10593* | The Tarski-Grothendieck axiom using abbreviations. This version is called Tarski's axiom: given a set 𝑥, there exists a set 𝑦 containing 𝑥, the subsets of the members of 𝑦, the power sets of the members of 𝑦, and the subsets of 𝑦 of cardinality less than that of 𝑦. (Contributed by NM, 21-Jun-2009.) |
| ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ 𝒫 𝑧 ∈ 𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≺ 𝑦 → 𝑧 ∈ 𝑦)) | ||
| Theorem | grothomex 10594 | The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9410). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.) |
| ⊢ ω ∈ V | ||
| Theorem | grothac 10595 | The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 10234). This can be put in a more conventional form via ween 9800 and dfac8 9900. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 9900). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.) |
| ⊢ dom card = V | ||
| Theorem | axgroth3 10596* | Alternate version of the Tarski-Grothendieck Axiom. ax-cc 10200 is used to derive this version. (Contributed by NM, 26-Mar-2007.) |
| ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → ((𝑦 ∖ 𝑧) ≼ 𝑧 ∨ 𝑧 ∈ 𝑦))) | ||
| Theorem | axgroth4 10597* | Alternate version of the Tarski-Grothendieck Axiom. ax-ac 10224 is used to derive this version. (Contributed by NM, 16-Apr-2007.) |
| ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∃𝑣 ∈ 𝑦 ∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ (𝑦 ∩ 𝑣)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → ((𝑦 ∖ 𝑧) ≼ 𝑧 ∨ 𝑧 ∈ 𝑦))) | ||
| Theorem | grothprimlem 10598* | Lemma for grothprim 10599. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.) |
| ⊢ ({𝑢, 𝑣} ∈ 𝑤 ↔ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣)))) | ||
| Theorem | grothprim 10599* | The Tarski-Grothendieck Axiom ax-groth 10588 expanded into set theory primitives using 163 symbols (allowing the defined symbols ∧, ∨, ↔, and ∃). An open problem is whether a shorter equivalent exists (when expanded to primitives). (Contributed by NM, 16-Apr-2007.) |
| ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧((𝑧 ∈ 𝑦 → ∃𝑣(𝑣 ∈ 𝑦 ∧ ∀𝑤(∀𝑢(𝑢 ∈ 𝑤 → 𝑢 ∈ 𝑧) → (𝑤 ∈ 𝑦 ∧ 𝑤 ∈ 𝑣)))) ∧ ∃𝑤((𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦) → (∀𝑣((𝑣 ∈ 𝑧 → ∃𝑡∀𝑢(∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑣 ∨ ℎ = 𝑢))) → 𝑢 = 𝑡)) ∧ (𝑣 ∈ 𝑦 → (𝑣 ∈ 𝑧 ∨ ∃𝑢(𝑢 ∈ 𝑧 ∧ ∃𝑔(𝑔 ∈ 𝑤 ∧ ∀ℎ(ℎ ∈ 𝑔 ↔ (ℎ = 𝑢 ∨ ℎ = 𝑣))))))) ∨ 𝑧 ∈ 𝑦)))) | ||
| Theorem | grothtsk 10600 | The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.) |
| ⊢ ∪ Tarski = V | ||
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