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Type | Label | Description |
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Statement | ||
Theorem | pwfseq 10701* | The powerset of a Dedekind-infinite set does not inject into the set of finite sequences. The proof is due to Halbeisen and Shelah. Proposition 1.7 of [KanamoriPincus] p. 418. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) | ||
Theorem | pwxpndom2 10702 | The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) (Proof shortened by AV, 18-Jul-2022.) |
⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ (𝐴 × 𝐴))) | ||
Theorem | pwxpndom 10703 | The powerset of a Dedekind-infinite set does not inject into its Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 × 𝐴)) | ||
Theorem | pwdjundom 10704 | The powerset of a Dedekind-infinite set does not inject into its cardinal sum with itself. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ (ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ (𝐴 ⊔ 𝐴)) | ||
Theorem | gchdjuidm 10705 | An infinite GCH-set is idempotent under cardinal sum. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ⊔ 𝐴) ≈ 𝐴) | ||
Theorem | gchxpidm 10706 | An infinite GCH-set is idempotent under cardinal product. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 × 𝐴) ≈ 𝐴) | ||
Theorem | gchpwdom 10707 | A relationship between dominance over the powerset and strict dominance when the sets involved are infinite GCH-sets. Proposition 3.1 of [KanamoriPincus] p. 421. (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝐵 ∈ GCH) → (𝐴 ≺ 𝐵 ↔ 𝒫 𝐴 ≼ 𝐵)) | ||
Theorem | gchaleph 10708 | If (ℵ‘𝐴) is a GCH-set and its powerset is well-orderable, then the successor aleph (ℵ‘suc 𝐴) is equinumerous to the powerset of (ℵ‘𝐴). (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴)) | ||
Theorem | gchaleph2 10709 | If (ℵ‘𝐴) and (ℵ‘suc 𝐴) are GCH-sets, then the successor aleph (ℵ‘suc 𝐴) is equinumerous to the powerset of (ℵ‘𝐴). (Contributed by Mario Carneiro, 31-May-2015.) |
⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴)) | ||
Theorem | hargch 10710 | If 𝐴 + ≈ 𝒫 𝐴, then 𝐴 is a GCH-set. The much simpler converse to gchhar 10716. (Contributed by Mario Carneiro, 2-Jun-2015.) |
⊢ ((har‘𝐴) ≈ 𝒫 𝐴 → 𝐴 ∈ GCH) | ||
Theorem | alephgch 10711 | If (ℵ‘suc 𝐴) is equinumerous to the powerset of (ℵ‘𝐴), then (ℵ‘𝐴) is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴) → (ℵ‘𝐴) ∈ GCH) | ||
Theorem | gch2 10712 | It is sufficient to require that all alephs are GCH-sets to ensure the full generalized continuum hypothesis. (The proof uses the Axiom of Regularity.) (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (GCH = V ↔ ran ℵ ⊆ GCH) | ||
Theorem | gch3 10713 | An equivalent formulation of the generalized continuum hypothesis. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (GCH = V ↔ ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) | ||
Theorem | gch-kn 10714* | 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 10618 to the successor aleph using enen2 9156. (Contributed by NM, 1-Oct-2004.) |
⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (ℵ‘suc 𝐴) ≈ (2o ↑m (ℵ‘𝐴)))) | ||
Theorem | gchaclem 10715 | Lemma for gchac 10718 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → ω ≼ 𝐴) & ⊢ (𝜑 → 𝒫 𝐶 ∈ GCH) & ⊢ (𝜑 → (𝐴 ≼ 𝐶 ∧ (𝐵 ≼ 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) ⇒ ⊢ (𝜑 → (𝐴 ≼ 𝒫 𝐶 ∧ (𝐵 ≼ 𝒫 𝒫 𝐶 → 𝒫 𝐴 ≼ 𝐵))) | ||
Theorem | gchhar 10716 | A "local" form of gchac 10718. 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 ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≈ 𝒫 𝐴) | ||
Theorem | gchacg 10717 | A "local" form of gchac 10718. 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 ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ∈ dom card) | ||
Theorem | gchac 10718 | 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) | ||
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 10860, 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 10721 and df-ina 10722 respectively ), Tarski classes (df-tsk 10786), and Grothendieck universes (df-gru 10828). We then introduce the Tarski's axiom ax-groth 10860 and prove various properties from that. | ||
Syntax | cwina 10719 | The class of weak inaccessibles. |
class Inaccw | ||
Syntax | cina 10720 | The class of strong inaccessibles. |
class Inacc | ||
Definition | df-wina 10721* | 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 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑥 𝑦 ≺ 𝑧)} | ||
Definition | df-ina 10722* | 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 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ≺ 𝑥)} | ||
Theorem | elwina 10723* | Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.) |
⊢ (𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | ||
Theorem | elina 10724* | Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.) |
⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴)) | ||
Theorem | winaon 10725 | A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On) | ||
Theorem | inawinalem 10726* | Lemma for inawina 10727. (Contributed by Mario Carneiro, 8-Jun-2014.) |
⊢ (𝐴 ∈ On → (∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) | ||
Theorem | inawina 10727 | Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw) | ||
Theorem | omina 10728 | ω is a strongly inaccessible cardinal. (Many definitions of "inaccessible" explicitly disallow ω as an inaccessible cardinal, but this choice allows to reuse our results for inaccessibles for ω.) (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ ω ∈ Inacc | ||
Theorem | winacard 10729 | A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴) | ||
Theorem | winainflem 10730* | A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) → ω ⊆ 𝐴) | ||
Theorem | winainf 10731 | A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴) | ||
Theorem | winalim 10732 | A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ (𝐴 ∈ Inaccw → Lim 𝐴) | ||
Theorem | winalim2 10733* | A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → ∃𝑥((ℵ‘𝑥) = 𝐴 ∧ Lim 𝑥)) | ||
Theorem | winafp 10734 | A nontrivial weakly inaccessible cardinal is a fixed point of the aleph function. (Contributed by Mario Carneiro, 29-May-2014.) |
⊢ ((𝐴 ∈ Inaccw ∧ 𝐴 ≠ ω) → (ℵ‘𝐴) = 𝐴) | ||
Theorem | winafpi 10735 | 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 4588 to turn this type of statement into the closed form statement winafp 10734, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 10734 using this theorem and dedth 4588, in ZFC. (You can prove this if you use ax-groth 10860, though.) (Contributed by Mario Carneiro, 28-May-2014.) |
⊢ 𝐴 ∈ Inaccw & ⊢ 𝐴 ≠ ω ⇒ ⊢ (ℵ‘𝐴) = 𝐴 | ||
Theorem | gchina 10736 | 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) | ||
Syntax | cwun 10737 | Extend class definition to include the class of all weak universes. |
class WUni | ||
Syntax | cwunm 10738 | Extend class definition to include the map whose value is the smallest weak universe of which the given set is a subset. |
class wUniCl | ||
Definition | df-wun 10739* | 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 10777) whereas the analogue for Grothendieck universes requires ax-groth 10860 (see grothtsk 10872). (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))} | ||
Definition | df-wunc 10740* | 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 ∣ 𝑥 ⊆ 𝑢}) | ||
Theorem | iswun 10741* | Properties of a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | ||
Theorem | wuntr 10742 | A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝑈 ∈ WUni → Tr 𝑈) | ||
Theorem | wununi 10743 | A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) | ||
Theorem | wunpw 10744 | A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈) | ||
Theorem | wunelss 10745 | The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝑈) | ||
Theorem | wunpr 10746 | A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) ⇒ ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) | ||
Theorem | wunun 10747 | A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) | ||
Theorem | wuntp 10748 | A weak universe is closed under unordered triple. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈) | ||
Theorem | wunss 10749 | A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑈) | ||
Theorem | wunin 10750 | A weak universe is closed under binary intersections. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑈) | ||
Theorem | wundif 10751 | A weak universe is closed under class difference. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝑈) | ||
Theorem | wunint 10752 | A weak universe is closed under nonempty intersections. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑈) | ||
Theorem | wunsn 10753 | A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → {𝐴} ∈ 𝑈) | ||
Theorem | wunsuc 10754 | A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → suc 𝐴 ∈ 𝑈) | ||
Theorem | wun0 10755 | A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) ⇒ ⊢ (𝜑 → ∅ ∈ 𝑈) | ||
Theorem | wunr1om 10756 | A weak universe is infinite, because it contains all the finite levels of the cumulative hierarchy. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) ⇒ ⊢ (𝜑 → (𝑅1 “ ω) ⊆ 𝑈) | ||
Theorem | wunom 10757 | A weak universe contains all the finite ordinals, and hence is infinite. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) ⇒ ⊢ (𝜑 → ω ⊆ 𝑈) | ||
Theorem | wunfi 10758 | A weak universe contains all finite sets with elements drawn from the universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ⊆ 𝑈) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝑈) | ||
Theorem | wunop 10759 | A weak universe is closed under ordered pairs. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑈) | ||
Theorem | wunot 10760 | A weak universe is closed under ordered triples. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐵, 𝐶〉 ∈ 𝑈) | ||
Theorem | wunxp 10761 | A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐴 × 𝐵) ∈ 𝑈) | ||
Theorem | wunpm 10762 | A weak universe is closed under partial mappings. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐴 ↑pm 𝐵) ∈ 𝑈) | ||
Theorem | wunmap 10763 | A weak universe is closed under mappings. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐴 ↑m 𝐵) ∈ 𝑈) | ||
Theorem | wunf 10764 | A weak universe is closed under functions with known domain and codomain. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝑈) | ||
Theorem | wundm 10765 | A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → dom 𝐴 ∈ 𝑈) | ||
Theorem | wunrn 10766 | A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) | ||
Theorem | wuncnv 10767 | A weak universe is closed under the converse operator. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → ◡𝐴 ∈ 𝑈) | ||
Theorem | wunres 10768 | A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐴 ↾ 𝐵) ∈ 𝑈) | ||
Theorem | wunfv 10769 | A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐴‘𝐵) ∈ 𝑈) | ||
Theorem | wunco 10770 | A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ 𝑈) | ||
Theorem | wuntpos 10771 | A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → tpos 𝐴 ∈ 𝑈) | ||
Theorem | intwun 10772 | The intersection of a collection of weak universes is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ ((𝐴 ⊆ WUni ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ WUni) | ||
Theorem | r1limwun 10773 | 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 10774 | 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 10775* | 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 10776* | Construct a weak universe from a given set. See also wunex2 10775. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑢 ∈ WUni 𝐴 ⊆ 𝑢) | ||
Theorem | uniwun 10777 | Every set is contained in a weak universe. This is the analogue of grothtsk 10872 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 10872. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ ∪ WUni = V | ||
Theorem | wunex3 10778 | Construct a weak universe from a given set. This version of wunex 10776 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 10779* | Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝐴 ∈ 𝑉 → (wUniCl‘𝐴) = ∩ {𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢}) | ||
Theorem | wuncid 10780 | The weak universe closure of a set contains the set. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (wUniCl‘𝐴)) | ||
Theorem | wunccl 10781 | 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 10782 | 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 10783 | The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝐴 ∈ 𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴)) | ||
Theorem | wuncval2 10784* | 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 10785 | Extend class definition to include the class of all Tarski classes. |
class Tarski | ||
Definition | df-tsk 10786* | 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 10860 and the equivalent axioms). Axiom A was first presented in Tarski's article Ueber unerreichbare Kardinalzahlen. Tarski introduced Axiom A to allow reasoning with inaccessible cardinals in ZFC. Later, Grothendieck introduced the concept of (Grothendieck) universes and showed they were exactly 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 10787* | 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 10788* | 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 10789 | 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 10790 | 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 10791 | 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 10792 | The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.) |
⊢ ∅ ∈ Tarski | ||
Theorem | tsksdom 10793 | 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 10794 | 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 10795 | 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 10796 | 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 10797 | 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 10798 | 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 10799 | 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 10800 | 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) |
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