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Theorem List for Metamath Proof Explorer - 10101-10200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgchdju1 10101 An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.)
((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ⊔ 1o) ≈ 𝐴)

Theoremgchinf 10102 An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝐴)

Theorempwfseqlem1 10103* Lemma for pwfseq 10109. Derive a contradiction by diagonalization. (Contributed by Mario Carneiro, 31-May-2015.)
(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥m 𝑛)–1-1𝑥)    &   𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})       ((𝜑𝜓) → 𝐷 ∈ ( 𝑛 ∈ ω (𝐴m 𝑛) ∖ 𝑛 ∈ ω (𝑥m 𝑛)))

Theorempwfseqlem2 10104* Lemma for pwfseq 10109. (Contributed by Mario Carneiro, 18-Nov-2014.) (Revised by AV, 18-Sep-2021.)
(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥m 𝑛)–1-1𝑥)    &   𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})    &   𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))       ((𝑌 ∈ Fin ∧ 𝑅𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))

Theorempwfseqlem3 10105* Lemma for pwfseq 10109. Using the construction 𝐷 from pwfseqlem1 10103, produce a function 𝐹 that maps any well-ordered infinite set to an element outside the set. (Contributed by Mario Carneiro, 31-May-2015.)
(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥m 𝑛)–1-1𝑥)    &   𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})    &   𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))       ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))

Theorempwfseqlem4a 10106* Lemma for pwfseqlem4 10107. (Contributed by Mario Carneiro, 7-Jun-2016.)
(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥m 𝑛)–1-1𝑥)    &   𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})    &   𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))       ((𝜑 ∧ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)

Theorempwfseqlem4 10107* Lemma for pwfseq 10109. Derive a final contradiction from the function 𝐹 in pwfseqlem3 10105. Applying fpwwe2 10088 to it, we get a certain maximal well-ordered subset 𝑍, but the defining property (𝑍𝐹(𝑊𝑍)) ∈ 𝑍 contradicts our assumption on 𝐹, so we are reduced to the case of 𝑍 finite. This too is a contradiction, though, because 𝑍 and its preimage under (𝑊𝑍) are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)
(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥m 𝑛)–1-1𝑥)    &   𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})    &   𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))    &   𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏𝑎 [(𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))}    &   𝑍 = dom 𝑊        ¬ 𝜑

Theorempwfseqlem5 10108* Lemma for pwfseq 10109. Although in some ways pwfseqlem4 10107 is the "main" part of the proof, one last aspect which makes up a remark in the original text is by far the hardest part to formalize. The main proof relies on the existence of an injection 𝐾 from the set of finite sequences on an infinite set 𝑥 to 𝑥. Now this alone would not be difficult to prove; this is mostly the claim of fseqen 9472. However, what is needed for the proof is a canonical injection on these sets, so we have to start from scratch pulling together explicit bijections from the lemmas.

If one attempts such a program, it will mostly go through, but there is one key step which is inherently nonconstructive, namely the proof of infxpen 9459. The resolution is not obvious, but it turns out that reversing an infinite ordinal's Cantor normal form absorbs all the non-leading terms (cnfcom3c 9187), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 9463). (Contributed by Mario Carneiro, 31-May-2015.)

(𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴m 𝑛))    &   (𝜑𝑋𝐴)    &   (𝜑𝐻:ω–1-1-onto𝑋)    &   (𝜓 ↔ ((𝑡𝐴𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))    &   (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))    &   𝑂 = OrdIso(𝑟, 𝑡)    &   𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂𝑢), (𝑂𝑣)⟩)    &   𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ 𝑇)    &   𝑆 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡m suc 𝑘) ↦ ((𝑓‘(𝑥𝑘))𝑃(𝑥𝑘)))), {⟨∅, (𝑂‘∅)⟩})    &   𝑄 = (𝑦 𝑛 ∈ ω (𝑡m 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)    &   𝐼 = (𝑥 ∈ ω, 𝑦𝑡 ↦ ⟨(𝑂𝑥), 𝑦⟩)    &   𝐾 = ((𝑃𝐼) ∘ 𝑄)        ¬ 𝜑

Theorempwfseq 10109* 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 (𝐴m 𝑛))

Theorempwxpndom2 10110 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)))

Theorempwxpndom 10111 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))

Theorempwdjundom 10112 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 ≼ (𝐴𝐴))

Theoremgchdjuidm 10113 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) → (𝐴𝐴) ≈ 𝐴)

Theoremgchxpidm 10114 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)

Theoremgchpwdom 10115 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 ≼ 𝐴𝐴 ∈ GCH ∧ 𝐵 ∈ GCH) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))

Theoremgchaleph 10116 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))

Theoremgchaleph2 10117 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))

Theoremhargch 10118 If 𝐴 + ≈ 𝒫 𝐴, then 𝐴 is a GCH-set. The much simpler converse to gchhar 10124. (Contributed by Mario Carneiro, 2-Jun-2015.)
((har‘𝐴) ≈ 𝒫 𝐴𝐴 ∈ GCH)

Theoremalephgch 10119 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)

Theoremgch2 10120 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)

Theoremgch3 10121 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))

Theoremgch-kn 10122* 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 10026 to the successor aleph using enen2 8672. (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 10123 Lemma for gchac 10126 (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 10124 A "local" form of gchac 10126. 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 10125 A "local" form of gchac 10126. 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 10126 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 10268, 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 10129 and df-ina 10130 respectively ), Tarski classes (df-tsk 10194), and Grothendieck universes (df-gru 10236). We then introduce the Tarski's axiom ax-groth 10268 and prove various properties from that.

4.1  Inaccessibles

4.1.1  Weakly and strongly inaccessible cardinals

Syntaxcwina 10127 The class of weak inaccessibles.
class Inaccw

Syntaxcina 10128 The class of strong inaccessibles.
class Inacc

Definitiondf-wina 10129* 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 10130* 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 10131* Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
(𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))

Theoremelina 10132* Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
(𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))

Theoremwinaon 10133 A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw𝐴 ∈ On)

Theoreminawinalem 10134* Lemma for inawina 10135. (Contributed by Mario Carneiro, 8-Jun-2014.)
(𝐴 ∈ On → (∀𝑥𝐴 𝒫 𝑥𝐴 → ∀𝑥𝐴𝑦𝐴 𝑥𝑦))

Theoreminawina 10135 Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)

Theoremomina 10136 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 10137 A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)

Theoremwinainflem 10138* A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
((𝐴 ≠ ∅ ∧ 𝐴 ∈ On ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → ω ⊆ 𝐴)

Theoremwinainf 10139 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → ω ⊆ 𝐴)

Theoremwinalim 10140 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → Lim 𝐴)

Theoremwinalim2 10141* 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 10142 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 10143 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 4471 to turn this type of statement into the closed form statement winafp 10142, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 10142 using this theorem and dedth 4471, in ZFC. (You can prove this if you use ax-groth 10268, though.) (Contributed by Mario Carneiro, 28-May-2014.)
𝐴 ∈ Inaccw    &   𝐴 ≠ ω       (ℵ‘𝐴) = 𝐴

Theoremgchina 10144 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 10145 Extend class definition to include the class of all weak universes.
class WUni

Syntaxcwunm 10146 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 10147* 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 10185) whereas the analogue for Grothendieck universes requires ax-groth 10268 (see grothtsk 10280). (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni = {𝑢 ∣ (Tr 𝑢𝑢 ≠ ∅ ∧ ∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢))}

Definitiondf-wunc 10148* 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 10149* 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 10150 A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈 ∈ WUni → Tr 𝑈)

Theoremwununi 10151 A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 𝐴𝑈)

Theoremwunpw 10152 A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → 𝒫 𝐴𝑈)

Theoremwunelss 10153 The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑𝐴𝑈)

Theoremwunpr 10154 A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Theoremwunun 10155 A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwuntp 10156 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 10157 A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝐴)       (𝜑𝐵𝑈)

Theoremwunin 10158 A weak universe is closed under binary intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwundif 10159 A weak universe is closed under class difference. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwunint 10160 A weak universe is closed under nonempty intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)

Theoremwunsn 10161 A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → {𝐴} ∈ 𝑈)

Theoremwunsuc 10162 A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → suc 𝐴𝑈)

Theoremwun0 10163 A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)       (𝜑 → ∅ ∈ 𝑈)

Theoremwunr1om 10164 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 10165 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 10166 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 10167 A weak universe is closed under ordered pairs. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Theoremwunot 10168 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 10169 A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)

Theoremwunpm 10170 A weak universe is closed under partial mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴pm 𝐵) ∈ 𝑈)

Theoremwunmap 10171 A weak universe is closed under mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴m 𝐵) ∈ 𝑈)

Theoremwunf 10172 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 10173 A weak universe is closed under the domain operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → dom 𝐴𝑈)

Theoremwunrn 10174 A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → ran 𝐴𝑈)

Theoremwuncnv 10175 A weak universe is closed under the converse operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑𝐴𝑈)

Theoremwunres 10176 A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwunfv 10177 A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwunco 10178 A weak universe is closed under composition. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)

Theoremwuntpos 10179 A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → tpos 𝐴𝑈)

Theoremintwun 10180 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 10181 Each limit stage in the cumulative hierarchy is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
((𝐴𝑉 ∧ Lim 𝐴) → (𝑅1𝐴) ∈ WUni)

Theoremr1wunlim 10182 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 10183* 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 10184* Construct a weak universe from a given set. See also wunex2 10183. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → ∃𝑢 ∈ WUni 𝐴𝑢)

Theoremuniwun 10185 Every set is contained in a weak universe. This is the analogue of grothtsk 10280 for weak universes, but it is provable in ZF without the Tarski-Grothendieck axiom, contrary to grothtsk 10280. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni = V

Theoremwunex3 10186 Construct a weak universe from a given set. This version of wunex 10184 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 10187* Value of the weak universe closure operator. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) = {𝑢 ∈ WUni ∣ 𝐴𝑢})

Theoremwuncid 10188 The weak universe closure of a set contains the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉𝐴 ⊆ (wUniCl‘𝐴))

Theoremwunccl 10189 The weak universe closure of a set is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘𝐴) ∈ WUni)

Theoremwuncss 10190 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 10191 The weak universe closure is idempotent. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴𝑉 → (wUniCl‘(wUniCl‘𝐴)) = (wUniCl‘𝐴))

Theoremwuncval2 10192* 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 10193 Extend class definition to include the class of all Tarski classes.
class Tarski

Definitiondf-tsk 10194* 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 10268 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 10195* 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 10196* 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 10197 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 10198 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 10199 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 10200 The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.)
∅ ∈ Tarski

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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45683
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