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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tskr1om2 10701 | A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 9571.) (Contributed by NM, 22-Feb-2011.) |
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ∪ (𝑅1 “ ω) ⊆ 𝑇) | ||
Theorem | tskinf 10702 | A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.) |
⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇) | ||
Theorem | tskpr 10703 | 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 10704 | 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 10705 | 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 10706 | A Tarski class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.) |
⊢ (𝑇 ∈ Tarski → 𝑇 ∈ dom card) | ||
Theorem | inttsk 10707 | 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 10708 | (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 10709 | Alternate proof of r1om 10177, shorter as a consequence of inar1 10708, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑅1‘ω) ≈ ω | ||
Theorem | rankcf 10710 | 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 10711 | (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 10712 | 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 10713 | 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 10714 | An even more direct relationship than r1tskina 10715 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 10715 | 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 10716 | 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 10717 | 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 10718 | 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 10719 | 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 10720 | 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 10721 | 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 10722 | 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 10723 | Extend class notation to include the class of all Grothendieck universes. |
class Univ | ||
Definition | df-gru 10724* | 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 10725* | 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 10726 | A Grothendieck universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝑈 ∈ Univ → Tr 𝑈) | ||
Theorem | gruelss 10727 | 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 10728 | 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 10729 | 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 10730 | 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 10731 | A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10732 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ∪ ran 𝐹 ∈ 𝑈) | ||
Theorem | gruiun 10732* | 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 10733 | A Grothendieck universe contains unions of its elements. (Contributed by Mario Carneiro, 17-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → ∪ 𝐴 ∈ 𝑈) | ||
Theorem | grurn 10734 | A Grothendieck universe contains the range of any function which takes values in the universe (see gruiun 10732 for a more intuitive version). (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐹:𝐴⟶𝑈) → ran 𝐹 ∈ 𝑈) | ||
Theorem | gruima 10735 | 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 10736 | 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 10737 | A Grothendieck universe contains the singletons of its elements. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈) → {𝐴} ∈ 𝑈) | ||
Theorem | gruop 10738 | 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 10739 | 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 10740 | 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 10741 | 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 10742* | 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 10743* | 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 10744 | 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 10745 | 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 10746 | A nonempty Grothendieck universe is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝑈 ∈ WUni) | ||
Theorem | intgru 10747 | The intersection of a family of universes is a universe. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ ((𝐴 ⊆ Univ ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ Univ) | ||
Theorem | ingru 10748* | 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 10749 | The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.) |
⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ) | ||
Theorem | grudomon 10750 | 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 10751 | 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 10752 | A characterization of Grothendieck universes, part 1. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ 𝐴 = (𝑈 ∩ On) ⇒ ⊢ (𝑈 ∈ Univ → (𝑅1‘𝐴) ⊆ 𝑈) | ||
Theorem | grur1 10753 | 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 10754 | Grothendieck universes are the same as transitive Tarski classes, part one: a transitive Tarski class is a universe. (The hard work is in tskuni 10716.) (Contributed by Mario Carneiro, 17-Jun-2013.) |
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇) → 𝑇 ∈ Univ) | ||
Theorem | grutsk 10755 | 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 10756* | 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 10767). 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 10757* | 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 10758* | 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 10759* | Derive the Axiom of Power Sets ax-pow 5319 from the Tarski-Grothendieck axiom ax-groth 10756. That it follows is mentioned by Bob Solovay at http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html 10756. Note that ax-pow 5319 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 10760 | Derive the Axiom of Power Sets from the Tarski-Grothendieck axiom ax-groth 10756. Note that ax-pow 5319 is not used by the proof. Use axpweq 5304 to obtain ax-pow 5319. Use pwex 5334 or pwexg 5332 instead. (Contributed by Gérard Lang, 22-Jun-2009.) (New usage is discouraged.) |
⊢ 𝒫 𝑥 ∈ V | ||
Theorem | axgroth6 10761* | 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 10762 | The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9576). 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 10763 | The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 10402). This can be put in a more conventional form via ween 9968 and dfac8 10068. 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 10068). (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.) |
⊢ dom card = V | ||
Theorem | axgroth3 10764* | Alternate version of the Tarski-Grothendieck Axiom. ax-cc 10368 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 10765* | Alternate version of the Tarski-Grothendieck Axiom. ax-ac 10392 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 10766* | Lemma for grothprim 10767. 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 10767* | The Tarski-Grothendieck Axiom ax-groth 10756 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 10768 | The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.) |
⊢ ∪ Tarski = V | ||
Theorem | inaprc 10769 | An equivalent to the Tarski-Grothendieck Axiom: there is a proper class of inaccessible cardinals. (Contributed by Mario Carneiro, 9-Jun-2013.) |
⊢ Inacc ∉ V | ||
Syntax | ctskm 10770 | Extend class definition to include the map whose value is the smallest Tarski class. |
class tarskiMap | ||
Definition | df-tskm 10771* | A function that maps a set 𝑥 to the smallest Tarski class that contains the set. (Contributed by FL, 30-Dec-2010.) |
⊢ tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ Tarski ∣ 𝑥 ∈ 𝑦}) | ||
Theorem | tskmval 10772* | Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.) |
⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥}) | ||
Theorem | tskmid 10773 | The set 𝐴 is an element of the smallest Tarski class that contains 𝐴. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴)) | ||
Theorem | tskmcl 10774 | A Tarski class that contains 𝐴 is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
⊢ (tarskiMap‘𝐴) ∈ Tarski | ||
Theorem | sstskm 10775* | Being a part of (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ⊆ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ⊆ 𝑥))) | ||
Theorem | eltskm 10776* | Belonging to (tarskiMap‘𝐴). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ (tarskiMap‘𝐴) ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥))) | ||
This section derives the basics of real and complex numbers. We first construct and axiomatize real and complex numbers (e.g., ax-resscn 11105). After that, we derive their basic properties, various operations like addition (df-add 11059) and sine (df-sin 15949), and subsets such as the integers (df-z 12497) and natural numbers (df-nn 12151). | ||
Syntax | cnpi 10777 |
The set of positive integers, which is the set of natural numbers ω
with 0 removed.
Note: This is the start of the Dedekind-cut construction of real and complex numbers. The last lemma of the construction is mulcnsrec 11077. The actual set of Dedekind cuts is defined by df-np 10914. |
class N | ||
Syntax | cpli 10778 | Positive integer addition. |
class +N | ||
Syntax | cmi 10779 | Positive integer multiplication. |
class ·N | ||
Syntax | clti 10780 | Positive integer ordering relation. |
class <N | ||
Syntax | cplpq 10781 | Positive pre-fraction addition. |
class +pQ | ||
Syntax | cmpq 10782 | Positive pre-fraction multiplication. |
class ·pQ | ||
Syntax | cltpq 10783 | Positive pre-fraction ordering relation. |
class <pQ | ||
Syntax | ceq 10784 | Equivalence class used to construct positive fractions. |
class ~Q | ||
Syntax | cnq 10785 | Set of positive fractions. |
class Q | ||
Syntax | c1q 10786 | The positive fraction constant 1. |
class 1Q | ||
Syntax | cerq 10787 | Positive fraction equivalence class. |
class [Q] | ||
Syntax | cplq 10788 | Positive fraction addition. |
class +Q | ||
Syntax | cmq 10789 | Positive fraction multiplication. |
class ·Q | ||
Syntax | crq 10790 | Positive fraction reciprocal operation. |
class *Q | ||
Syntax | cltq 10791 | Positive fraction ordering relation. |
class <Q | ||
Syntax | cnp 10792 | Set of positive reals. |
class P | ||
Syntax | c1p 10793 | Positive real constant 1. |
class 1P | ||
Syntax | cpp 10794 | Positive real addition. |
class +P | ||
Syntax | cmp 10795 | Positive real multiplication. |
class ·P | ||
Syntax | cltp 10796 | Positive real ordering relation. |
class <P | ||
Syntax | cer 10797 | Equivalence class used to construct signed reals. |
class ~R | ||
Syntax | cnr 10798 | Set of signed reals. |
class R | ||
Syntax | c0r 10799 | The signed real constant 0. |
class 0R | ||
Syntax | c1r 10800 | The signed real constant 1. |
class 1R |
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