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Theorem List for Metamath Proof Explorer - 9701-9800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgch-kn 9701* The equivalence of two versions of the Generalized Continuum Hypothesis. The right-hand side is the standard version in the literature. The left-hand side is a version devised by Kannan Nambiar, which he calls the Axiom of Combinatorial Sets. For the notation and motivation behind this axiom, see his paper, "Derivation of Continuum Hypothesis from Axiom of Combinatorial Sets," available at http://www.e-atheneum.net/science/derivation_ch.pdf. The equivalence of the two sides provides a negative answer to Open Problem 2 in http://www.e-atheneum.net/science/open_problem_print.pdf. The key idea in the proof below is to equate both sides of alephexp2 9605 to the successor aleph using enen2 8257. (Contributed by NM, 1-Oct-2004.)
(𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (ℵ‘suc 𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))))
 
3.4.2  Derivation of the Axiom of Choice
 
Theoremgchaclem 9702 Lemma for gchac 9705 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑 → ω ≼ 𝐴)    &   (𝜑 → 𝒫 𝐶 ∈ GCH)    &   (𝜑 → (𝐴𝐶 ∧ (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴𝐵)))       (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴𝐵)))
 
Theoremgchhar 9703 A "local" form of gchac 9705. If 𝐴 and 𝒫 𝐴 are GCH-sets, then the Hartogs number of 𝐴 is 𝒫 𝐴 (so 𝒫 𝐴 and a fortiori 𝐴 are well-orderable). The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
((ω ≼ 𝐴𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≈ 𝒫 𝐴)
 
Theoremgchacg 9704 A "local" form of gchac 9705. If 𝐴 and 𝒫 𝐴 are GCH-sets, then 𝒫 𝐴 is well-orderable. The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 15-May-2015.)
((ω ≼ 𝐴𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ∈ dom card)
 
Theoremgchac 9705 The Generalized Continuum Hypothesis implies the Axiom of Choice. The original proof is due to Sierpiński (1947); we use a refinement of Sierpiński's result due to Specker. (Contributed by Mario Carneiro, 15-May-2015.)
(GCH = V → CHOICE)
 
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY

Here we introduce Tarski-Grothendieck (TG) set theory, named after mathematicians Alfred Tarski and Alexander Grothendieck. TG theory extends ZFC with the TG Axiom ax-groth 9847, which states that 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."

We first introduce the concept of inaccessibles, including weakly and strongly inaccessible cardinals (df-wina 9708 and df-ina 9709 respectively ), Tarski classes (df-tsk 9773), and Grothendieck universes (df-gru 9815). We then introduce the Tarski's axiom ax-groth 9847 and prove various properties from that.

 
4.1  Inaccessibles
 
4.1.1  Weakly and strongly inaccessible cardinals
 
Syntaxcwina 9706 The class of weak inaccessibles.
class Inaccw
 
Syntaxcina 9707 The class of strong inaccessibles.
class Inacc
 
Definitiondf-wina 9708* An ordinal is weakly inaccessible iff it is a regular limit cardinal. Note that our definition allows ω as a weakly inaccessible cardinal. (Contributed by Mario Carneiro, 22-Jun-2013.)
Inaccw = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 𝑦𝑧)}
 
Definitiondf-ina 9709* An ordinal is strongly inaccessible iff it is a regular strong limit cardinal, which is to say that it dominates the powersets of every smaller ordinal. (Contributed by Mario Carneiro, 22-Jun-2013.)
Inacc = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥 𝒫 𝑦𝑥)}
 
Theoremelwina 9710* Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
(𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
 
Theoremelina 9711* Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
(𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
 
Theoremwinaon 9712 A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw𝐴 ∈ On)
 
Theoreminawinalem 9713* Lemma for inawina 9714. (Contributed by Mario Carneiro, 8-Jun-2014.)
(𝐴 ∈ On → (∀𝑥𝐴 𝒫 𝑥𝐴 → ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
 
Theoreminawina 9714 Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
 
Theoremomina 9715 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 9716 A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
 
Theoremwinainflem 9717* A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
((𝐴 ≠ ∅ ∧ 𝐴 ∈ On ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → ω ⊆ 𝐴)
 
Theoremwinainf 9718 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → ω ⊆ 𝐴)
 
Theoremwinalim 9719 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → Lim 𝐴)
 
Theoremwinalim2 9720* 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 9721 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 9722 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 4278 to turn this type of statement into the closed form statement winafp 9721, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 9721 using this theorem and dedth 4278, in ZFC. (You can prove this if you use ax-groth 9847, though.) (Contributed by Mario Carneiro, 28-May-2014.)
𝐴 ∈ Inaccw    &   𝐴 ≠ ω       (ℵ‘𝐴) = 𝐴
 
Theoremgchina 9723 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 9724 Extend class definition to include the class of all weak universes.
class WUni
 
Syntaxcwunm 9725 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 9726* 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 9764) whereas the analogue for Grothendieck universes requires ax-groth 9847 (see grothtsk 9859). (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni = {𝑢 ∣ (Tr 𝑢𝑢 ≠ ∅ ∧ ∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢))}
 
Definitiondf-wunc 9727* 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 9728* 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 9729 A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈 ∈ WUni → Tr 𝑈)
 
Theoremwununi 9730 A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 𝐴𝑈)
 
Theoremwunpw 9731 A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → 𝒫 𝐴𝑈)
 
Theoremwunelss 9732 The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑𝐴𝑈)
 
Theoremwunpr 9733 A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremwunun 9734 A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwuntp 9735 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 9736 A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝐴)       (𝜑𝐵𝑈)
 
Theoremwunin 9737 A weak universe is closed under binary intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwundif 9738 A weak universe is closed under class difference. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwunint 9739 A weak universe is closed under nonempty intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
 
Theoremwunsn 9740 A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → {𝐴} ∈ 𝑈)
 
Theoremwunsuc 9741 A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → suc 𝐴𝑈)
 
Theoremwun0 9742 A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)       (𝜑 → ∅ ∈ 𝑈)
 
Theoremwunr1om 9743 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 9744 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 9745 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 9746 A weak universe is closed under ordered pairs. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
 
Theoremwunot 9747 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 9748 A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)
 
Theoremwunpm 9749 A weak universe is closed under partial mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴pm 𝐵) ∈ 𝑈)
 
Theoremwunmap 9750 A weak universe is closed under mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝑚 𝐵) ∈ 𝑈)
 
Theoremwunf 9751 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 9752 A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → dom 𝐴𝑈)
 
Theoremwunrn 9753 A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → ran 𝐴𝑈)
 
Theoremwuncnv 9754 A weak universe is closed under the converse operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑𝐴𝑈)
 
Theoremwunres 9755 A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwunfv 9756 A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwunco 9757 A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwuntpos 9758 A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → tpos 𝐴𝑈)
 
Theoremintwun 9759 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 9760 Each limit stage in the cumulative hierarchy is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)
 
Theoremr1wunlim 9761 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 9762* 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 9763* Construct a weak universe from a given set. See also wunex2 9762. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → ∃𝑢 ∈ WUni 𝐴𝑢)
 
Theoremuniwun 9764 Every set is contained in a weak universe. This is the analogue of grothtsk 9859 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 9859. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni = V
 
Theoremwunex3 9765 Construct a weak universe from a given set. This version of wunex 9763 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 9766* Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) = {𝑢 ∈ WUni ∣ 𝐴𝑢})
 
Theoremwuncid 9767 The weak universe closure of a set contains the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉𝐴 ⊆ (wUniCl‘𝐴))
 
Theoremwunccl 9768 The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) ∈ WUni)
 
Theoremwuncss 9769 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 9770 The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴))
 
Theoremwuncval2 9771* 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 9772 Extend class definition to include the class of all Tarski classes.
class Tarski
 
Definitiondf-tsk 9773* 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 9847 and the equivalent axioms). Axiom A was first presented in Tarski's article _Über 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 9774* 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 9775* 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 9776 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 9777 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 9778 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 9779 The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.)
∅ ∈ Tarski
 
Theoremtsksdom 9780 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 9781 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 9782 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 9783 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 9784 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 9785 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 9786 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 9787 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 9788 One is an element of a nonempty Tarski class. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 1𝑜𝑇)
 
Theoremtsk2 9789 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 9790 If a Tarski class is not empty, it has more than two elements. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → 2𝑜𝑇)
 
Theoremtskr1om 9791 A nonempty Tarski class is infinite, because it contains all the finite levels of the cumulative hierarchy. (This proof does not use ax-inf 8699.) (Contributed by Mario Carneiro, 24-Jun-2013.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
 
Theoremtskr1om2 9792 A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 8699.) (Contributed by NM, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (𝑅1 “ ω) ⊆ 𝑇)
 
Theoremtskinf 9793 A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → ω ≼ 𝑇)
 
Theoremtskpr 9794 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 9795 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 9796 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 9797 A Tarski class is well-orderable. (Contributed by Mario Carneiro, 20-Jun-2013.)
(𝑇 ∈ Tarski → 𝑇 ∈ dom card)
 
Theoreminttsk 9798 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 9799 (𝑅1𝐴) for 𝐴 a strongly inaccessible cardinal is equipotent to 𝐴. (Contributed by Mario Carneiro, 6-Jun-2013.)
(𝐴 ∈ Inacc → (𝑅1𝐴) ≈ 𝐴)
 
Theoremr1omALT 9800 Alternate proof of r1om 9268, shorter as a consequence of inar1 9799, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑅1‘ω) ≈ ω
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