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Theorem List for Metamath Proof Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgchi 9201 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 ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)
 
Theoremgchen1 9202 If 𝐴𝐵 < 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) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)
 
Theoremgchen2 9203 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) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≈ 𝒫 𝐴)
 
Theoremgchor 9204 If 𝐴𝐵 ≤ 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) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → (𝐴𝐵𝐵 ≈ 𝒫 𝐴))
 
Theoremengch 9205 The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.)
(𝐴𝐵 → (𝐴 ∈ GCH ↔ 𝐵 ∈ GCH))
 
Theoremgchdomtri 9206 Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 9258. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴 ∈ GCH ∧ (𝐴 +𝑐 𝐴) ≈ 𝐴𝐵 ≼ 𝒫 𝐴) → (𝐴𝐵𝐵𝐴))
 
Theoremfpwwe2cbv 9207* Lemma for fpwwe2 9220. (Contributed by Mario Carneiro, 3-Jun-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}       𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}
 
Theoremfpwwe2lem1 9208* Lemma for fpwwe2 9220. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}       𝑊 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
 
Theoremfpwwe2lem2 9209* Lemma for fpwwe2 9220. (Contributed by Mario Carneiro, 19-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)       (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
 
Theoremfpwwe2lem3 9210* Lemma for fpwwe2 9220. (Contributed by Mario Carneiro, 19-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑋𝑊𝑅)       ((𝜑𝐵𝑋) → ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵)
 
Theoremfpwwe2lem5 9211* Lemma for fpwwe2 9220. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)       ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)
 
Theoremfpwwe2lem6 9212* Lemma for fpwwe2 9220. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑𝐵 ∈ dom 𝑀)    &   (𝜑𝐵 ∈ dom 𝑁)    &   (𝜑 → (𝑀𝐵) = (𝑁𝐵))       ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑋𝐶𝑌 ∧ (𝑀𝐶) = (𝑁𝐶)))
 
Theoremfpwwe2lem7 9213* Lemma for fpwwe2 9220. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑𝐵 ∈ dom 𝑀)    &   (𝜑𝐵 ∈ dom 𝑁)    &   (𝜑 → (𝑀𝐵) = (𝑁𝐵))       ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑆(𝑁𝐵) ∧ (𝐷𝑅(𝑀𝐵) → (𝐶𝑅𝐷𝐶𝑆𝐷))))
 
Theoremfpwwe2lem8 9214* Lemma for fpwwe2 9220. Show by induction that the two isometries 𝑀 and 𝑁 agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑 → dom 𝑀 ⊆ dom 𝑁)       (𝜑𝑀 = (𝑁 ↾ dom 𝑀))
 
Theoremfpwwe2lem9 9215* Lemma for fpwwe2 9220. Given two well-orders 𝑋, 𝑅 and 𝑌, 𝑆 of parts of 𝐴, one is an initial segment of the other. (The 𝑂𝑃 hypothesis is in order to break the symmetry of 𝑋 and 𝑌.) (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑 → dom 𝑀 ⊆ dom 𝑁)       (𝜑 → (𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))
 
Theoremfpwwe2lem10 9216* Lemma for fpwwe2 9220. 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.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)       (𝜑 → ((𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))
 
Theoremfpwwe2lem11 9217* Lemma for fpwwe2 9220. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
 
Theoremfpwwe2lem12 9218* Lemma for fpwwe2 9220. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑𝑋 ∈ dom 𝑊)
 
Theoremfpwwe2lem13 9219* Lemma for fpwwe2 9220. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑 → (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)
 
Theoremfpwwe2 9220* Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset 𝑋, (𝑊𝑋)⟩ 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 8612. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
 
Theoremfpwwecbv 9221* Lemma for fpwwe 9223. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}       𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))}
 
Theoremfpwwelem 9222* Lemma for fpwwe 9223. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}    &   (𝜑𝐴 ∈ V)       (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))))
 
Theoremfpwwe 9223* Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 7874 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset 𝑋, (𝑊𝑋)⟩ 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 8612. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
 
Theoremcanth4 9224* An "effective" form of Cantor's theorem canth 6385. 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⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}    &   𝐵 = dom 𝑊    &   𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})       ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
 
Theoremcanthnumlem 9225* Lemma for canthnum 9226. (Contributed by Mario Carneiro, 19-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}    &   𝐵 = dom 𝑊    &   𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})       (𝐴𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
 
Theoremcanthnum 9226 The set of well-orderable subsets of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 7874. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)
(𝐴𝑉𝐴 ≺ (𝒫 𝐴 ∩ dom card))
 
Theoremcanthwelem 9227* Lemma for canthnum 9226. (Contributed by Mario Carneiro, 31-May-2015.)
𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}    &   𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   𝐵 = dom 𝑊    &   𝐶 = ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})       (𝐴𝑉 → ¬ 𝐹:𝑂1-1𝐴)
 
Theoremcanthwe 9228* The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 7874. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)
𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}       (𝐴𝑉𝐴𝑂)
 
Theoremcanthp1lem1 9229 Lemma for canthp1 9231. (Contributed by Mario Carneiro, 18-May-2015.)
(1𝑜𝐴 → (𝐴 +𝑐 2𝑜) ≼ 𝒫 𝐴)
 
Theoremcanthp1lem2 9230* Lemma for canthp1 9231. (Contributed by Mario Carneiro, 18-May-2015.)
(𝜑 → 1𝑜𝐴)    &   (𝜑𝐹:𝒫 𝐴1-1-onto→(𝐴 +𝑐 1𝑜))    &   (𝜑𝐺:((𝐴 +𝑐 1𝑜) ∖ {(𝐹𝐴)})–1-1-onto𝐴)    &   𝐻 = ((𝐺𝐹) ∘ (𝑥 ∈ 𝒫 𝐴 ↦ if(𝑥 = 𝐴, ∅, 𝑥)))    &   𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐻‘(𝑟 “ {𝑦})) = 𝑦))}    &   𝐵 = dom 𝑊        ¬ 𝜑
 
Theoremcanthp1 9231 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.)
(1𝑜𝐴 → (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴)
 
Theoremfinngch 9232 The exclusion of finite sets from consideration in df-gch 9198 is necessary, because otherwise finite sets larger than a singleton would violate the GCH property. (Contributed by Mario Carneiro, 10-Jun-2015.)
((𝐴 ∈ Fin ∧ 1𝑜𝐴) → (𝐴 ≺ (𝐴 +𝑐 1𝑜) ∧ (𝐴 +𝑐 1𝑜) ≺ 𝒫 𝐴))
 
Theoremgchcda1 9233 An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.)
((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 +𝑐 1𝑜) ≈ 𝐴)
 
Theoremgchinf 9234 An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝐴)
 
Theorempwfseqlem1 9235* Lemma for pwfseq 9241. 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 9236* Lemma for pwfseq 9241. (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 9237* Lemma for pwfseq 9241. Using the construction 𝐷 from pwfseqlem1 9235, 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 9238* Lemma for pwfseqlem4 9239. (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 9239* Lemma for pwfseq 9241. Derive a final contradiction from the function 𝐹 in pwfseqlem3 9237. Applying fpwwe2 9220 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 9240* Lemma for pwfseq 9241. Although in some ways pwfseqlem4 9239 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 8609. 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 8596. 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 8362), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 8600). (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 9241* 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 9242 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 +𝑐 (𝐴 × 𝐴)))
 
Theorempwxpndom 9243 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))
 
Theorempwcdandom 9244 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 +𝑐 𝐴))
 
Theoremgchcdaidm 9245 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 +𝑐 𝐴) ≈ 𝐴)
 
Theoremgchxpidm 9246 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 9247 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 9248 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 9249 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 9250 If 𝐴 + ≈ 𝒫 𝐴, then 𝐴 is a GCH-set. The much simpler converse to gchhar 9256. (Contributed by Mario Carneiro, 2-Jun-2015.)
((har‘𝐴) ≈ 𝒫 𝐴𝐴 ∈ GCH)
 
Theoremalephgch 9251 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 9252 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 9253 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 9254* 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 9158 to the successor aleph using enen2 7862. (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 9255 Lemma for gchac 9258 (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 9256 A "local" form of gchac 9258. 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 9257 A "local" form of gchac 9258. 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 9258 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 9400, 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 9261 and df-ina 9262 respectively ), Tarski classes (df-tsk 9326), and Grothendieck universes (df-gru 9368). We then introduce the Tarski's axiom ax-groth 9400 and prove various properties from that.

 
4.1  Inaccessibles
 
4.1.1  Weakly and strongly inaccessible cardinals
 
Syntaxcwina 9259 The class of weak inaccessibles.
class Inaccw
 
Syntaxcina 9260 The class of strong inaccessibles.
class Inacc
 
Definitiondf-wina 9261* 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 9262* 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 9263* Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
(𝐴 ∈ Inaccw ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
 
Theoremelina 9264* Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)
(𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥𝐴 𝒫 𝑥𝐴))
 
Theoremwinaon 9265 A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw𝐴 ∈ On)
 
Theoreminawinalem 9266* Lemma for inawina 9267. (Contributed by Mario Carneiro, 8-Jun-2014.)
(𝐴 ∈ On → (∀𝑥𝐴 𝒫 𝑥𝐴 → ∀𝑥𝐴𝑦𝐴 𝑥𝑦))
 
Theoreminawina 9267 Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
 
Theoremomina 9268 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 9269 A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → (card‘𝐴) = 𝐴)
 
Theoremwinainflem 9270* A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
((𝐴 ≠ ∅ ∧ 𝐴 ∈ On ∧ ∀𝑥𝐴𝑦𝐴 𝑥𝑦) → ω ⊆ 𝐴)
 
Theoremwinainf 9271 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → ω ⊆ 𝐴)
 
Theoremwinalim 9272 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)
(𝐴 ∈ Inaccw → Lim 𝐴)
 
Theoremwinalim2 9273* 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 9274 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 9275 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 3992 to turn this type of statement into the closed form statement winafp 9274, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 9274 using this theorem and dedth 3992, in ZFC. (You can prove this if you use ax-groth 9400, though.) (Contributed by Mario Carneiro, 28-May-2014.)
𝐴 ∈ Inaccw    &   𝐴 ≠ ω       (ℵ‘𝐴) = 𝐴
 
Theoremgchina 9276 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 9277 Extend class definition to include the class of all weak universes.
class WUni
 
Syntaxcwunm 9278 Extend class definition to include the map whose value is the smallest weak universe.
class wUniCl
 
Definitiondf-wun 9279* The class of all weak universes. A weak universe is a nonempty transitive class closed under union, pairing, and powerset. The advantage of a weak universe over a Grothendieck universe is that weak universes satisfy the analogue uniwun 9317 of grothtsk 9412 in ZFC (whereas grothtsk 9412 requires ax-groth 9400). (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni = {𝑢 ∣ (Tr 𝑢𝑢 ≠ ∅ ∧ ∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢))}
 
Definitiondf-wunc 9280* 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 9281* 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 9282 A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝑈 ∈ WUni → Tr 𝑈)
 
Theoremwununi 9283 A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 𝐴𝑈)
 
Theoremwunpw 9284 A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → 𝒫 𝐴𝑈)
 
Theoremwunelss 9285 The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑𝐴𝑈)
 
Theoremwunpr 9286 A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
 
Theoremwunun 9287 A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwuntp 9288 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 9289 A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝐴)       (𝜑𝐵𝑈)
 
Theoremwunin 9290 A weak universe is closed under intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwundif 9291 A weak universe is closed under class difference. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → (𝐴𝐵) ∈ 𝑈)
 
Theoremwunint 9292 A weak universe is closed under intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       ((𝜑𝐴 ≠ ∅) → 𝐴𝑈)
 
Theoremwunsn 9293 A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → {𝐴} ∈ 𝑈)
 
Theoremwunsuc 9294 A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)       (𝜑 → suc 𝐴𝑈)
 
Theoremwun0 9295 A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)       (𝜑 → ∅ ∈ 𝑈)
 
Theoremwunr1om 9296 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 9297 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 9298 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 9299 A weak universe is closed under ordered pairs. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝑈 ∈ WUni)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)       (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
 
Theoremwunot 9300 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)
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