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Theorem List for Metamath Proof Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreminawina 10101 Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
 
Theoremomina 10102 ω is a strongly inaccessible cardinal. (Many definitions of "inaccessible" explicitly disallow ω as an inaccessible cardinal, but this choice allows us to reuse our results for inaccessibles for ω.) (Contributed by Mario Carneiro, 29-May-2014.)
ω ∈ Inacc
 
Theoremwinacard 10103 A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
 
Theoremwinainflem 10104* A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
((𝐴 ≠ ∅ ∧ 𝐴 ∈ On ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → ω ⊆ 𝐴)
 
Theoremwinainf 10105 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → ω ⊆ 𝐴)
 
Theoremwinalim 10106 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → Lim 𝐴)
 
Theoremwinalim2 10107* A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)
((𝐴 ∈ Inaccw𝐴 ≠ ω) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥))
 
Theoremwinafp 10108 A nontrivial weakly inaccessible cardinal is a fixed point of the aleph function. (Contributed by Mario Carneiro, 29-May-2014.)
((𝐴 ∈ Inaccw𝐴 ≠ ω) → (ℵ‘𝐴) = 𝐴)
 
Theoremwinafpi 10109 This theorem, which states that a nontrivial inaccessible cardinal is its own aleph number, is stated here in inference form, where the assumptions are in the hypotheses rather than an antecedent. Often, we use dedth 4521 to turn this type of statement into the closed form statement winafp 10108, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 10108 using this theorem and dedth 4521, in ZFC. (You can prove this if you use ax-groth 10234, though.) (Contributed by Mario Carneiro, 28-May-2014.)
𝐴 ∈ Inaccw    &   𝐴 ≠ ω       (ℵ‘𝐴) = 𝐴
 
Theoremgchina 10110 Assuming the GCH, weakly and strongly inaccessible cardinals coincide. Theorem 11.20 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 5-Jun-2015.)
(GCH = V → Inaccw = Inacc)
 
4.1.2  Weak universes
 
Syntaxcwun 10111 Extend class definition to include the class of all weak universes.
class WUni
 
Syntaxcwunm 10112 Extend class definition to include the map whose value is the smallest weak universe of which the given set is a subset.
class wUniCl
 
Definitiondf-wun 10113* The class of all weak universes. A weak universe is a nonempty transitive class closed under union, pairing, and powerset. The advantage of weak universes over Grothendieck universes is that one can prove that every set is contained in a weak universe in ZF (see uniwun 10151) whereas the analogue for Grothendieck universes requires ax-groth 10234 (see grothtsk 10246). (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni = {𝑢 ∣ (Tr 𝑢𝑢 ≠ ∅ ∧ ∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢))}
 
Definitiondf-wunc 10114* A function that maps a set 𝑥 to the smallest weak universe that contains the elements of the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
wUniCl = (𝑥 ∈ V ↦ {𝑢 ∈ WUni ∣ 𝑥𝑢})
 
Theoremiswun 10115* Properties of a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈𝑉 → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
 
Theoremwuntr 10116 A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈 ∈ WUni → Tr 𝑈)
 
Theoremwununi 10117 A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 𝐴𝑈)
 
Theoremwunpw 10118 A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → 𝒫 𝐴𝑈)
 
Theoremwunelss 10119 The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑𝐴𝑈)
 
Theoremwunpr 10120 A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremwunun 10121 A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwuntp 10122 A weak universe is closed under unordered triple. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑𝐶𝑈)       (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)
 
Theoremwunss 10123 A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝐴)       (𝜑𝐵𝑈)
 
Theoremwunin 10124 A weak universe is closed under binary intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwundif 10125 A weak universe is closed under class difference. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwunint 10126 A weak universe is closed under nonempty intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
 
Theoremwunsn 10127 A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → {𝐴} ∈ 𝑈)
 
Theoremwunsuc 10128 A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → suc 𝐴𝑈)
 
Theoremwun0 10129 A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)       (𝜑 → ∅ ∈ 𝑈)
 
Theoremwunr1om 10130 A weak universe is infinite, because it contains all the finite levels of the cumulative hierarchy. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)       (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)
 
Theoremwunom 10131 A weak universe contains all the finite ordinals, and hence is infinite. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)       (𝜑 → ω ⊆ 𝑈)
 
Theoremwunfi 10132 A weak universe contains all finite sets with elements drawn from the universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐴 ∈ Fin)       (𝜑𝐴𝑈)
 
Theoremwunop 10133 A weak universe is closed under ordered pairs. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
 
Theoremwunot 10134 A weak universe is closed under ordered triples. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑𝐶𝑈)       (𝜑 → ⟨𝐴, 𝐵, 𝐶⟩ ∈ 𝑈)
 
Theoremwunxp 10135 A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)
 
Theoremwunpm 10136 A weak universe is closed under partial mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴pm 𝐵) ∈ 𝑈)
 
Theoremwunmap 10137 A weak universe is closed under mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴m 𝐵) ∈ 𝑈)
 
Theoremwunf 10138 A weak universe is closed under functions with known domain and codomain. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑𝐹:𝐴𝐵)       (𝜑𝐹𝑈)
 
Theoremwundm 10139 A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → dom 𝐴𝑈)
 
Theoremwunrn 10140 A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → ran 𝐴𝑈)
 
Theoremwuncnv 10141 A weak universe is closed under the converse operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑𝐴𝑈)
 
Theoremwunres 10142 A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwunfv 10143 A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwunco 10144 A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwuntpos 10145 A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → tpos 𝐴𝑈)
 
Theoremintwun 10146 The intersection of a collection of weak universes is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝐴 ⊆ WUni ∧ 𝐴 ≠ ∅) → 𝐴 ∈ WUni)
 
Theoremr1limwun 10147 Each limit stage in the cumulative hierarchy is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
 
Theoremr1wunlim 10148 The weak universes in the cumulative hierarchy are exactly the limit ordinals. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → ((𝑅1𝐴) ∈ WUni ↔ Lim 𝐴))
 
Theoremwunex2 10149* Construct a weak universe from a given set. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 𝑧) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)    &   𝑈 = ran 𝐹       (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
 
Theoremwunex 10150* Construct a weak universe from a given set. See also wunex2 10149. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → ∃𝑢 ∈ WUni 𝐴𝑢)
 
Theoremuniwun 10151 Every set is contained in a weak universe. This is the analogue of grothtsk 10246 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 10246. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni = V
 
Theoremwunex3 10152 Construct a weak universe from a given set. This version of wunex 10150 has a simpler proof, but requires the axiom of regularity. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝑈 = (𝑅1‘((rank‘𝐴) +o ω))       (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
 
Theoremwuncval 10153* Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) = {𝑢 ∈ WUni ∣ 𝐴𝑢})
 
Theoremwuncid 10154 The weak universe closure of a set contains the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉𝐴 ⊆ (wUniCl‘𝐴))
 
Theoremwunccl 10155 The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) ∈ WUni)
 
Theoremwuncss 10156 The weak universe closure is a subset of any other weak universe containing the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝑈 ∈ WUni ∧ 𝐴𝑈) → (wUniCl‘𝐴) ⊆ 𝑈)
 
Theoremwuncidm 10157 The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴))
 
Theoremwuncval2 10158* 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) ∪ 𝑥𝑧 ({𝒫 𝑥, 𝑥} ∪ ran (𝑦𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1o)) ↾ ω)    &   𝑈 = ran 𝐹       (𝐴𝑉 → (wUniCl‘𝐴) = 𝑈)
 
4.1.3  Tarski classes
 
Syntaxctsk 10159 Extend class definition to include the class of all Tarski classes.
class Tarski
 
Definitiondf-tsk 10160* 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 10234 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 ∣ (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ ∃𝑤𝑦 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))}
 
Theoremeltskg 10161* Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.)
(𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ ∃𝑤𝑇 𝒫 𝑧𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
 
Theoremeltsk2g 10162* Properties of a Tarski class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
(𝑇𝑉 → (𝑇 ∈ Tarski ↔ (∀𝑧𝑇 (𝒫 𝑧𝑇 ∧ 𝒫 𝑧𝑇) ∧ ∀𝑧 ∈ 𝒫 𝑇(𝑧𝑇𝑧𝑇))))
 
Theoremtskpwss 10163 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 ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
 
Theoremtskpw 10164 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 ∧ 𝐴𝑇) → 𝒫 𝐴𝑇)
 
Theoremtsken 10165 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 ∧ 𝐴𝑇) → (𝐴𝑇𝐴𝑇))
 
Theorem0tsk 10166 The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.)
∅ ∈ Tarski
 
Theoremtsksdom 10167 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 ∧ 𝐴𝑇) → 𝐴𝑇)
 
Theoremtskssel 10168 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 ∧ 𝐴𝑇𝐴𝑇) → 𝐴𝑇)
 
Theoremtskss 10169 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 ∧ 𝐴𝑇𝐵𝐴) → 𝐵𝑇)
 
Theoremtskin 10170 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 ∧ 𝐴𝑇) → (𝐴𝐵) ∈ 𝑇)
 
Theoremtsksn 10171 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 ∧ 𝐴𝑇) → {𝐴} ∈ 𝑇)
 
Theoremtsktrss 10172 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 𝐴𝐴𝑇) → 𝐴𝑇)
 
Theoremtsksuc 10173 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 ∧ 𝐴𝑇) → suc 𝐴𝑇)
 
Theoremtsk0 10174 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)
 
Theoremtsk1 10175 One is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1o𝑇)
 
Theoremtsk2 10176 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 ≠ ∅) → 2o𝑇)
 
Theorem2domtsk 10177 If a Tarski class is not empty, it has more than two elements. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2o𝑇)
 
Theoremtskr1om 10178 A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 9090.) (Contributed by Mario Carneiro, 24-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
 
Theoremtskr1om2 10179 A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 9090.) (Contributed by NM, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
 
Theoremtskinf 10180 A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇)
 
Theoremtskpr 10181 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 ∧ 𝐴𝑇𝐵𝑇) → {𝐴, 𝐵} ∈ 𝑇)
 
Theoremtskop 10182 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 ∧ 𝐴𝑇𝐵𝑇) → ⟨𝐴, 𝐵⟩ ∈ 𝑇)
 
Theoremtskxpss 10183 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 ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ⊆ 𝑇)
 
Theoremtskwe2 10184 A Tarski class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)
(𝑇 ∈ Tarski → 𝑇 ∈ dom card)
 
Theoreminttsk 10185 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)
 
Theoreminar1 10186 (𝑅1𝐴) for 𝐴 a strongly inaccessible cardinal is equipotent to 𝐴. (Contributed by Mario Carneiro, 6-Jun-2013.)
(𝐴 ∈ Inacc → (𝑅1𝐴) ≈ 𝐴)
 
Theoremr1omALT 10187 Alternate proof of r1om 9655, shorter as a consequence of inar1 10186, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑅1‘ω) ≈ ω
 
Theoremrankcf 10188 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))
 
Theoreminatsk 10189 (𝑅1𝐴) for 𝐴 a strongly inaccessible cardinal is a Tarski class. (Contributed by Mario Carneiro, 8-Jun-2013.)
(𝐴 ∈ Inacc → (𝑅1𝐴) ∈ Tarski)
 
Theoremr1omtsk 10190 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
 
Theoremtskord 10191 A Tarski class contains all ordinals smaller than it. (Contributed by Mario Carneiro, 8-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝐴 ∈ On ∧ 𝐴𝑇) → 𝐴𝑇)
 
Theoremtskcard 10192 An even more direct relationship than r1tskina 10193 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)
 
Theoremr1tskina 10193 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 → ((𝑅1𝐴) ∈ Tarski ↔ (𝐴 = ∅ ∨ 𝐴 ∈ Inacc)))
 
Theoremtskuni 10194 The union of an element of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 22-Jun-2013.)
((𝑇 ∈ Tarski ∧ Tr 𝑇𝐴𝑇) → 𝐴𝑇)
 
Theoremtskwun 10195 A nonempty transitive Tarski class is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝑇 ∈ Tarski ∧ Tr 𝑇𝑇 ≠ ∅) → 𝑇 ∈ WUni)
 
Theoremtskint 10196 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) ∧ 𝐴𝑇𝐴 ≠ ∅) → 𝐴𝑇)
 
Theoremtskun 10197 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) ∧ 𝐴𝑇𝐵𝑇) → (𝐴𝐵) ∈ 𝑇)
 
Theoremtskxp 10198 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) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 × 𝐵) ∈ 𝑇)
 
Theoremtskmap 10199 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) ∧ 𝐴𝑇𝐵𝑇) → (𝐴m 𝐵) ∈ 𝑇)
 
Theoremtskurn 10200 A transitive Tarski class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013.)
(((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
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