P. 107 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10601-10700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	zfcndreg 10601*	Axiom of Regularity ax-reg 9553, reproved from conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2402. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
Theorem	zfcndinf 10602*	Axiom of Infinity ax-inf 9606, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing Theorem el 5419 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))
Theorem	zfcndac 10603*	Axiom of Choice ax-ac 10442, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣))
3.4  The Generalized Continuum Hypothesis
3.4.1  Sets satisfying the Generalized Continuum Hypothesis
Syntax	cgch 10604	Extend class notation to include the collection of sets that satisfy the GCH.
class GCH
Definition	df-gch 10605*	Define the collection of "GCH-sets", or sets for which the generalized continuum hypothesis holds. In this language the generalized continuum hypothesis can be expressed as GCH = V. A set 𝑥 satisfies the generalized continuum hypothesis if it is finite or there is no set 𝑦 strictly between 𝑥 and its powerset in cardinality. The continuum hypothesis is equivalent to ω ∈ GCH. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)})
Theorem	elgch 10606*	Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴 ≺ 𝑥 ∧ 𝑥 ≺ 𝒫 𝐴))))
Theorem	fingch 10607	A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ Fin ⊆ GCH
Theorem	gchi 10608	The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ ((𝐴 ∈ GCH ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
Theorem	gchen1 10609	If 𝐴 ≤ 𝐵 < 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then 𝐴 = 𝐵 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝒫 𝐴)) → 𝐴 ≈ 𝐵)
Theorem	gchen2 10610	If 𝐴 < 𝐵 ≤ 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then 𝐵 = 𝒫 𝐴 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≈ 𝒫 𝐴)
Theorem	gchor 10611	If 𝐴 ≤ 𝐵 ≤ 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then either 𝐴 = 𝐵 or 𝐵 = 𝒫 𝐴 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝒫 𝐴)) → (𝐴 ≈ 𝐵 ∨ 𝐵 ≈ 𝒫 𝐴))
Theorem	engch 10612	The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ GCH ↔ 𝐵 ∈ GCH))
Theorem	gchdomtri 10613	Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 10665. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ ((𝐴 ∈ GCH ∧ (𝐴 ⊔ 𝐴) ≈ 𝐴 ∧ 𝐵 ≼ 𝒫 𝐴) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴))
Theorem	fpwwe2cbv 10614*	Lemma for fpwwe2 10627. (Contributed by Mario Carneiro, 3-Jun-2015.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    ⇒   ⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}
Theorem	fpwwe2lem1 10615*	Lemma for fpwwe2 10627. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    ⇒   ⊢ 𝑊 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
Theorem	fpwwe2lem2 10616*	Lemma for fpwwe2 10627. (Contributed by Mario Carneiro, 19-May-2015.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
Theorem	fpwwe2lem3 10617*	Lemma for fpwwe2 10627. (Contributed by Mario Carneiro, 19-May-2015.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋𝑊𝑅)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵)
Theorem	fpwwe2lem4 10618*	Lemma for fpwwe2 10627. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.) (Proof shortened by Matthew House, 10-Sep-2025.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    ⇒   ⊢ ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)
Theorem	fpwwe2lem5 10619*	Lemma for fpwwe2 10627. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑊𝑅)    &   ⊢ (𝜑 → 𝑌𝑊𝑆)    &   ⊢ 𝑀 = OrdIso(𝑅, 𝑋)    &   ⊢ 𝑁 = OrdIso(𝑆, 𝑌)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑀)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑁)    &   ⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))    ⇒   ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ (◡𝑀‘𝐶) = (◡𝑁‘𝐶)))
Theorem	fpwwe2lem6 10620*	Lemma for fpwwe2 10627. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑊𝑅)    &   ⊢ (𝜑 → 𝑌𝑊𝑆)    &   ⊢ 𝑀 = OrdIso(𝑅, 𝑋)    &   ⊢ 𝑁 = OrdIso(𝑆, 𝑌)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑀)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑁)    &   ⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))    ⇒   ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶𝑆(𝑁‘𝐵) ∧ (𝐷𝑅(𝑀‘𝐵) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷))))
Theorem	fpwwe2lem7 10621*	Lemma for fpwwe2 10627. Show by induction that the two isometries 𝑀 and 𝑁 agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑊𝑅)    &   ⊢ (𝜑 → 𝑌𝑊𝑆)    &   ⊢ 𝑀 = OrdIso(𝑅, 𝑋)    &   ⊢ 𝑁 = OrdIso(𝑆, 𝑌)    &   ⊢ (𝜑 → dom 𝑀 ⊆ dom 𝑁)    ⇒   ⊢ (𝜑 → 𝑀 = (𝑁 ↾ dom 𝑀))
Theorem	fpwwe2lem8 10622*	Lemma for fpwwe2 10627. Given two well-orders ⟨𝑋, 𝑅⟩ and ⟨𝑌, 𝑆⟩ of parts of 𝐴, one is an initial segment of the other. (The 𝑂 ⊆ 𝑃 hypothesis is in order to break the symmetry of 𝑋 and 𝑌.) (Contributed by Mario Carneiro, 15-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑊𝑅)    &   ⊢ (𝜑 → 𝑌𝑊𝑆)    &   ⊢ 𝑀 = OrdIso(𝑅, 𝑋)    &   ⊢ 𝑁 = OrdIso(𝑆, 𝑌)    &   ⊢ (𝜑 → dom 𝑀 ⊆ dom 𝑁)    ⇒   ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))
Theorem	fpwwe2lem9 10623*	Lemma for fpwwe2 10627. Given two well-orders ⟨𝑋, 𝑅⟩ and ⟨𝑌, 𝑆⟩ of parts of 𝐴, one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑊𝑅)    &   ⊢ (𝜑 → 𝑌𝑊𝑆)    ⇒   ⊢ (𝜑 → ((𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))
Theorem	fpwwe2lem10 10624*	Lemma for fpwwe2 10627. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   ⊢ 𝑋 = ∪ dom 𝑊    ⇒   ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
Theorem	fpwwe2lem11 10625*	Lemma for fpwwe2 10627. (Contributed by Mario Carneiro, 18-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   ⊢ 𝑋 = ∪ dom 𝑊    ⇒   ⊢ (𝜑 → 𝑋 ∈ dom 𝑊)
Theorem	fpwwe2lem12 10626*	Lemma for fpwwe2 10627. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   ⊢ 𝑋 = ∪ dom 𝑊    ⇒   ⊢ (𝜑 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)
Theorem	fpwwe2 10627*	Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset ⟨𝑋, (𝑊‘𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 10013. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   ⊢ 𝑋 = ∪ dom 𝑊    ⇒   ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))
Theorem	fpwwecbv 10628*	Lemma for fpwwe 10630. (Contributed by Mario Carneiro, 15-May-2015.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}    ⇒   ⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (𝐹‘(◡𝑠 “ {𝑧})) = 𝑧))}
Theorem	fpwwelem 10629*	Lemma for fpwwe 10630. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))))
Theorem	fpwwe 10630*	Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 9117 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset ⟨𝑋, (𝑊‘𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 10013. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹‘𝑥) ∈ 𝐴)    &   ⊢ 𝑋 = ∪ dom 𝑊    ⇒   ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹‘𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))
Theorem	canth4 10631*	An "effective" form of Cantor's theorem canth 7364. For any function 𝐹 from the powerset of 𝐴 to 𝐴, there are two definable sets 𝐵 and 𝐶 which witness non-injectivity of 𝐹. Corollary 1.3 of [KanamoriPincus] p. 416. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}    &   ⊢ 𝐵 = ∪ dom 𝑊    &   ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)})    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))
Theorem	canthnumlem 10632*	Lemma for canthnum 10633. (Contributed by Mario Carneiro, 19-May-2015.)
⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}    &   ⊢ 𝐵 = ∪ dom 𝑊    &   ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)})    ⇒   ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
Theorem	canthnum 10633	The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9117. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ (𝒫 𝐴 ∩ dom card))
Theorem	canthwelem 10634*	Lemma for canthwe 10635. (Contributed by Mario Carneiro, 31-May-2015.)
⊢ 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}    &   ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   ⊢ 𝐵 = ∪ dom 𝑊    &   ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})    ⇒   ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:𝑂–1-1→𝐴)
Theorem	canthwe 10635*	The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9117. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)
⊢ 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝑂)
Theorem	canthp1lem1 10636	Lemma for canthp1 10638. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 2o) ≼ 𝒫 𝐴)
Theorem	canthp1lem2 10637*	Lemma for canthp1 10638. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ (𝜑 → 1o ≺ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝒫 𝐴–1-1-onto→(𝐴 ⊔ 1o))    &   ⊢ (𝜑 → 𝐺:((𝐴 ⊔ 1o) ∖ {(𝐹‘𝐴)})–1-1-onto→𝐴)    &   ⊢ 𝐻 = ((𝐺 ∘ 𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))    &   ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐻‘(◡𝑟 “ {𝑦})) = 𝑦))}    &   ⊢ 𝐵 = ∪ dom 𝑊    ⇒   ⊢ ¬ 𝜑
Theorem	canthp1 10638	A slightly stronger form of Cantor's theorem: For 1 < 𝑛, 𝑛 + 1 < 2↑𝑛. Corollary 1.6 of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ (1o ≺ 𝐴 → (𝐴 ⊔ 1o) ≺ 𝒫 𝐴)
Theorem	finngch 10639	The exclusion of finite sets from consideration in df-gch 10605 is necessary, because otherwise finite sets larger than a singleton would violate the GCH property. (Contributed by Mario Carneiro, 10-Jun-2015.)
⊢ ((𝐴 ∈ Fin ∧ 1o ≺ 𝐴) → (𝐴 ≺ (𝐴 ⊔ 1o) ∧ (𝐴 ⊔ 1o) ≺ 𝒫 𝐴))
Theorem	gchdju1 10640	An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 ⊔ 1o) ≈ 𝐴)
Theorem	gchinf 10641	An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝐴)
Theorem	pwfseqlem1 10642*	Lemma for pwfseq 10648. Derive a contradiction by diagonalization. (Contributed by Mario Carneiro, 31-May-2015.)
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)    &   ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)    &   ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ (∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∖ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)))
Theorem	pwfseqlem2 10643*	Lemma for pwfseq 10648. (Contributed by Mario Carneiro, 18-Nov-2014.) (Revised by AV, 18-Sep-2021.)
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)    &   ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)    &   ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})    &   ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))    ⇒   ⊢ ((𝑌 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑌𝐹𝑅) = (𝐻‘(card‘𝑌)))
Theorem	pwfseqlem3 10644*	Lemma for pwfseq 10648. Using the construction 𝐷 from pwfseqlem1 10642, produce a function 𝐹 that maps any well-ordered infinite set to an element outside the set. (Contributed by Mario Carneiro, 31-May-2015.)
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)    &   ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)    &   ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})    &   ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))
Theorem	pwfseqlem4a 10645*	Lemma for pwfseqlem4 10646. (Contributed by Mario Carneiro, 7-Jun-2016.)
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)    &   ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)    &   ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})    &   ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))    ⇒   ⊢ ((𝜑 ∧ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎) ∧ 𝑠 We 𝑎)) → (𝑎𝐹𝑠) ∈ 𝐴)
Theorem	pwfseqlem4 10646*	Lemma for pwfseq 10648. Derive a final contradiction from the function 𝐹 in pwfseqlem3 10644. Applying fpwwe2 10627 to it, we get a certain maximal well-ordered subset 𝑍, but the defining property (𝑍𝐹(𝑊‘𝑍)) ∈ 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 (𝑊‘𝑍) 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.) (Proof shortened by Matthew House, 10-Sep-2025.)
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)    &   ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))    &   ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)    &   ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})    &   ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))    &   ⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑏 ∈ 𝑎 [(◡𝑠 “ {𝑏}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑏))}    &   ⊢ 𝑍 = ∪ dom 𝑊    ⇒   ⊢ ¬ 𝜑
Theorem	pwfseqlem5 10647*	Lemma for pwfseq 10648. Although in some ways pwfseqlem4 10646 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 10010. 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 9997. 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 9674), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 10001). (Contributed by Mario Carneiro, 31-May-2015.)
⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)    &   ⊢ (𝜓 ↔ ((𝑡 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑡 × 𝑡) ∧ 𝑟 We 𝑡) ∧ ω ≼ 𝑡))    &   ⊢ (𝜑 → ∀𝑏 ∈ (har‘𝒫 𝐴)(ω ⊆ 𝑏 → (𝑁‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))    &   ⊢ 𝑂 = OrdIso(𝑟, 𝑡)    &   ⊢ 𝑇 = (𝑢 ∈ dom 𝑂, 𝑣 ∈ dom 𝑂 ↦ ⟨(𝑂‘𝑢), (𝑂‘𝑣)⟩)    &   ⊢ 𝑃 = ((𝑂 ∘ (𝑁‘dom 𝑂)) ∘ ◡𝑇)    &   ⊢ 𝑆 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝑡 ↑m suc 𝑘) ↦ ((𝑓‘(𝑥 ↾ 𝑘))𝑃(𝑥‘𝑘)))), {⟨∅, (𝑂‘∅)⟩})    &   ⊢ 𝑄 = (𝑦 ∈ ∪ 𝑛 ∈ ω (𝑡 ↑m 𝑛) ↦ ⟨dom 𝑦, ((𝑆‘dom 𝑦)‘𝑦)⟩)    &   ⊢ 𝐼 = (𝑥 ∈ ω, 𝑦 ∈ 𝑡 ↦ ⟨(𝑂‘𝑥), 𝑦⟩)    &   ⊢ 𝐾 = ((𝑃 ∘ 𝐼) ∘ 𝑄)    ⇒   ⊢ ¬ 𝜑
Theorem	pwfseq 10648*	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 10649	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 10650	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 10651	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 10652	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 10653	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 10654	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 10655	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 10656	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 10657	If 𝐴 + ≈ 𝒫 𝐴, then 𝐴 is a GCH-set. The much simpler converse to gchhar 10663. (Contributed by Mario Carneiro, 2-Jun-2015.)
⊢ ((har‘𝐴) ≈ 𝒫 𝐴 → 𝐴 ∈ GCH)
Theorem	alephgch 10658	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 10659	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 10660	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 10661*	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 10565 to the successor aleph using enen2 9105. (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))))
3.4.2  Derivation of the Axiom of Choice
Theorem	gchaclem 10662	Lemma for gchac 10665 (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 10663	A "local" form of gchac 10665. 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 10664	A "local" form of gchac 10665. 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 10665	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 10807, 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 10668 and df-ina 10669 respectively ), Tarski classes (df-tsk 10733), and Grothendieck universes (df-gru 10775). We then introduce the Tarski's axiom ax-groth 10807 and prove various properties from that.
4.1  Inaccessibles
4.1.1  Weakly and strongly inaccessible cardinals
Syntax	cwina 10666	The class of weak inaccessibles.
class Inaccw
Syntax	cina 10667	The class of strong inaccessibles.
class Inacc
Definition	df-wina 10668*	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 10669*	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 10670*	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 10671*	Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
Theorem	winaon 10672	A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On)
Theorem	inawinalem 10673*	Lemma for inawina 10674. (Contributed by Mario Carneiro, 8-Jun-2014.)
⊢ (𝐴 ∈ On → (∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦))
Theorem	inawina 10674	Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
Theorem	omina 10675	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 10676	A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
⊢ (𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
Theorem	winainflem 10677*	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 10678	A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
⊢ (𝐴 ∈ Inaccw → ω ⊆ 𝐴)
Theorem	winalim 10679	A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
⊢ (𝐴 ∈ Inaccw → Lim 𝐴)
Theorem	winalim2 10680*	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 10681	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 10682	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 4545 to turn this type of statement into the closed form statement winafp 10681, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 10681 using this theorem and dedth 4545, in ZFC. (You can prove this if you use ax-groth 10807, though.) (Contributed by Mario Carneiro, 28-May-2014.)
⊢ 𝐴 ∈ Inaccw    &   ⊢ 𝐴 ≠ ω    ⇒   ⊢ (ℵ‘𝐴) = 𝐴
Theorem	gchina 10683	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
Syntax	cwun 10684	Extend class definition to include the class of all weak universes.
class WUni
Syntax	cwunm 10685	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 10686*	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 10724) whereas the analogue for Grothendieck universes requires ax-groth 10807 (see grothtsk 10819). (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 10687*	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 10688*	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 10689	A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝑈 ∈ WUni → Tr 𝑈)
Theorem	wununi 10690	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 10691	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 10692	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 10693	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 10694	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 10695	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 10696	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 10697	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 10698	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 10699	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 10700	A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    ⇒   ⊢ (𝜑 → {𝐴} ∈ 𝑈)