Home | Metamath
Proof Explorer Theorem List (p. 93 of 460) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28853) |
Hilbert Space Explorer
(28854-30376) |
Users' Mathboxes
(30377-45962) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cantnfvalf 9201* | Lemma for cantnf 9229. The function appearing in cantnfval 9204 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.) |
⊢ 𝐹 = seqω((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)), ∅) ⇒ ⊢ 𝐹:ω⟶On | ||
Theorem | cantnfs 9202 | Elementhood in the set of finitely supported functions from 𝐵 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) ⇒ ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))) | ||
Theorem | cantnfcl 9203 | Basic properties of the order isomorphism 𝐺 used later. The support of an 𝐹 ∈ 𝑆 is a finite subset of 𝐴, so it is well-ordered by E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) ⇒ ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω)) | ||
Theorem | cantnfval 9204* | The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺)) | ||
Theorem | cantnfval2 9205* | Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)‘dom 𝐺)) | ||
Theorem | cantnfsuc 9206* | The value of the recursive function 𝐻 at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾))) | ||
Theorem | cantnfle 9207* | A lower bound on the CNF function. Since ((𝐴 CNF 𝐵)‘𝐹) is defined as the sum of (𝐴 ↑o 𝑥) ·o (𝐹‘𝑥) over all 𝑥 in the support of 𝐹, it is larger than any of these terms (and all other terms are zero, so we can extend the statement to all 𝐶 ∈ 𝐵 instead of just those 𝐶 in the support). (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 28-Jun-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐴 ↑o 𝐶) ·o (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹)) | ||
Theorem | cantnflt 9208* | An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent 𝐴 ↑o 𝐶 where 𝐶 is larger than any exponent (𝐺‘𝑥), 𝑥 ∈ 𝐾 which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅) & ⊢ (𝜑 → ∅ ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ suc dom 𝐺) & ⊢ (𝜑 → 𝐶 ∈ On) & ⊢ (𝜑 → (𝐺 “ 𝐾) ⊆ 𝐶) ⇒ ⊢ (𝜑 → (𝐻‘𝐾) ∈ (𝐴 ↑o 𝐶)) | ||
Theorem | cantnflt2 9209 | An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → ∅ ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ On) & ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐶) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶)) | ||
Theorem | cantnff 9210 | The CNF function is a function from finitely supported functions from 𝐵 to 𝐴, to the ordinal exponential 𝐴 ↑o 𝐵. (Contributed by Mario Carneiro, 28-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) ⇒ ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵)) | ||
Theorem | cantnf0 9211 | The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → ∅ ∈ 𝐴) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅) | ||
Theorem | cantnfrescl 9212* | A function is finitely supported from 𝐵 to 𝐴 iff the extended function is finitely supported from 𝐷 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → 𝐵 ⊆ 𝐷) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅) & ⊢ (𝜑 → ∅ ∈ 𝐴) & ⊢ 𝑇 = dom (𝐴 CNF 𝐷) ⇒ ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇)) | ||
Theorem | cantnfres 9213* | The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → 𝐵 ⊆ 𝐷) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅) & ⊢ (𝜑 → ∅ ∈ 𝐴) & ⊢ 𝑇 = dom (𝐴 CNF 𝐷) & ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋))) | ||
Theorem | cantnfp1lem1 9214* | Lemma for cantnfp1 9217. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋) & ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝑆) | ||
Theorem | cantnfp1lem2 9215* | Lemma for cantnfp1 9217. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋) & ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) & ⊢ (𝜑 → ∅ ∈ 𝑌) & ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅)) ⇒ ⊢ (𝜑 → dom 𝑂 = suc ∪ dom 𝑂) | ||
Theorem | cantnfp1lem3 9216* | Lemma for cantnfp1 9217. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋) & ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) & ⊢ (𝜑 → ∅ ∈ 𝑌) & ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐹‘(𝑂‘𝑘))) +o 𝑧)), ∅) & ⊢ 𝐾 = OrdIso( E , (𝐺 supp ∅)) & ⊢ 𝑀 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐾‘𝑘)) ·o (𝐺‘(𝐾‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))) | ||
Theorem | cantnfp1 9217* | If 𝐹 is created by adding a single term (𝐹‘𝑋) = 𝑌 to 𝐺, where 𝑋 is larger than any element of the support of 𝐺, then 𝐹 is also a finitely supported function and it is assigned the value ((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑧 where 𝑧 is the value of 𝐺. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋) & ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) ⇒ ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))) | ||
Theorem | oemapso 9218* | The relation 𝑇 is a strict order on 𝑆 (a corollary of wemapso2 9090). (Contributed by Mario Carneiro, 28-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ (𝜑 → 𝑇 Or 𝑆) | ||
Theorem | oemapval 9219* | Value of the relation 𝑇. (Contributed by Mario Carneiro, 28-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) | ||
Theorem | oemapvali 9220* | If 𝐹 < 𝐺, then there is some 𝑧 witnessing this, but we can say more and in fact there is a definable expression 𝑋 that also witnesses 𝐹 < 𝐺. (Contributed by Mario Carneiro, 25-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))) | ||
Theorem | cantnflem1a 9221* | Lemma for cantnf 9229. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅)) | ||
Theorem | cantnflem1b 9222* | Lemma for cantnf 9229. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} & ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅)) ⇒ ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂‘𝑢)) | ||
Theorem | cantnflem1c 9223* | Lemma for cantnf 9229. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} & ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅)) ⇒ ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅)) | ||
Theorem | cantnflem1d 9224* | Lemma for cantnf 9229. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} & ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ (𝐻‘suc (◡𝑂‘𝑋))) | ||
Theorem | cantnflem1 9225* | Lemma for cantnf 9229. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation 𝑇 is a strict order, but we haven't shown it is a well-order yet. But being a strict order is enough to show that two distinct 𝐹, 𝐺 are 𝑇 -related as 𝐹 < 𝐺 or 𝐺 < 𝐹, and WLOG assuming that 𝐹 < 𝐺, we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 2-Jul-2019.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐹 ∈ 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐹𝑇𝐺) & ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} & ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅) ⇒ ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘𝐺)) | ||
Theorem | cantnflem2 9226* | Lemma for cantnf 9229. (Contributed by Mario Carneiro, 28-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵)) & ⊢ (𝜑 → ∅ ∈ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o))) | ||
Theorem | cantnflem3 9227* | Lemma for cantnf 9229. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than 𝐶 has a normal form, we can use oeeu 8260 to factor 𝐶 into the form ((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴 ↑o 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴 ↑o 𝑋) ≤ (𝐴 ↑o 𝑋) ·o 𝑌 ≤ 𝐶, 𝑍 has a normal form, and by appending the term (𝐴 ↑o 𝑋) ·o 𝑌 using cantnfp1 9217 we get a normal form for 𝐶. (Contributed by Mario Carneiro, 28-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵)) & ⊢ (𝜑 → ∅ ∈ 𝐶) & ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)} & ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = 〈𝑎, 𝑏〉 ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶)) & ⊢ 𝑌 = (1st ‘𝑃) & ⊢ 𝑍 = (2nd ‘𝑃) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍) & ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) ⇒ ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵)) | ||
Theorem | cantnflem4 9228* | Lemma for cantnf 9229. Complete the induction step of cantnflem3 9227. (Contributed by Mario Carneiro, 25-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵)) & ⊢ (𝜑 → ∅ ∈ 𝐶) & ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)} & ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = 〈𝑎, 𝑏〉 ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶)) & ⊢ 𝑌 = (1st ‘𝑃) & ⊢ 𝑍 = (2nd ‘𝑃) ⇒ ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵)) | ||
Theorem | cantnf 9229* | The Cantor Normal Form theorem. The function (𝐴 CNF 𝐵), which maps a finitely supported function from 𝐵 to 𝐴 to the sum ((𝐴 ↑o 𝑓(𝑎1)) ∘ 𝑎1) +o ((𝐴 ↑o 𝑓(𝑎2)) ∘ 𝑎2) +o ... over all indices 𝑎 < 𝐵 such that 𝑓(𝑎) is nonzero, is an order isomorphism from the ordering 𝑇 of finitely supported functions to the set (𝐴 ↑o 𝐵) under the natural order. Setting 𝐴 = ω and letting 𝐵 be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 9213, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵))) | ||
Theorem | oemapwe 9230* | The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternate definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵))) | ||
Theorem | cantnffval2 9231* | An alternate definition of df-cnf 9198 which relies on cantnf 9229. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9200 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) | ||
Theorem | cantnff1o 9232 | Simplify the isomorphism of cantnf 9229 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ 𝑆 = dom (𝐴 CNF 𝐵) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) ⇒ ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵)) | ||
Theorem | wemapwe 9233* | Construct lexicographic order on a function space based on a reverse well-ordering of the indices and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by AV, 3-Jul-2019.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑆 We 𝐵) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ 𝐹 = OrdIso(𝑅, 𝐴) & ⊢ 𝐺 = OrdIso(𝑆, 𝐵) & ⊢ 𝑍 = (𝐺‘∅) ⇒ ⊢ (𝜑 → 𝑇 We 𝑈) | ||
Theorem | oef1o 9234* | A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7070.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶) & ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐴 ∈ (On ∖ 1o)) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐶 ∈ On) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → (𝐹‘∅) = ∅) & ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))) & ⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)) ⇒ ⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷)) | ||
Theorem | cnfcomlem 9235* | Lemma for cnfcom 9236. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ (𝜑 → 𝐼 ∈ dom 𝐺) & ⊢ (𝜑 → 𝑂 ∈ (ω ↑o (𝐺‘𝐼))) & ⊢ (𝜑 → (𝑇‘𝐼):(𝐻‘𝐼)–1-1-onto→𝑂) ⇒ ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) | ||
Theorem | cnfcom 9236* | Any ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ (𝜑 → 𝐼 ∈ dom 𝐺) ⇒ ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼)))) | ||
Theorem | cnfcom2lem 9237* | Lemma for cnfcom2 9238. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ 𝑊 = (𝐺‘∪ dom 𝐺) & ⊢ (𝜑 → ∅ ∈ 𝐵) ⇒ ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺) | ||
Theorem | cnfcom2 9238* | Any nonzero ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) |
⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ 𝑊 = (𝐺‘∪ dom 𝐺) & ⊢ (𝜑 → ∅ ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊))) | ||
Theorem | cnfcom3lem 9239* | Lemma for cnfcom3 9240. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.) |
⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ 𝑊 = (𝐺‘∪ dom 𝐺) & ⊢ (𝜑 → ω ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o)) | ||
Theorem | cnfcom3 9240* | Any infinite ordinal 𝐵 is equinumerous to a power of ω. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 9242.) (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 4-Jul-2019.) |
⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴)) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ 𝑊 = (𝐺‘∪ dom 𝐺) & ⊢ (𝜑 → ω ⊆ 𝐵) & ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢)) & ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣)) & ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)) ⇒ ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑o 𝑊)) | ||
Theorem | cnfcom3clem 9241* | Lemma for cnfcom3c 9242. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.) |
⊢ 𝑆 = dom (ω CNF 𝐴) & ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝑏) & ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) & ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) & ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅) & ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) & ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥))) & ⊢ 𝑊 = (𝐺‘∪ dom 𝐺) & ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢)) & ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣)) & ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)) & ⊢ 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁) ⇒ ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) | ||
Theorem | cnfcom3c 9242* | Wrap the construction of cnfcom3 9240 into an existential quantifier. For any ω ⊆ 𝑏, there is a bijection from 𝑏 to some power of ω. Furthermore, this bijection is canonical , which means that we can find a single function 𝑔 which will give such bijections for every 𝑏 less than some arbitrarily large bound 𝐴. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) | ||
Theorem | trcl 9243* | For any set 𝐴, show the properties of its transitive closure 𝐶. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 9244 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω) & ⊢ 𝐶 = ∪ 𝑦 ∈ ω (𝐹‘𝑦) ⇒ ⊢ (𝐴 ⊆ 𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥)) | ||
Theorem | tz9.1 9244* |
Every set has a transitive closure (the smallest transitive extension).
Theorem 9.1 of [TakeutiZaring] p.
73. See trcl 9243 for an explicit
expression for the transitive closure. Apparently open problems are
whether this theorem can be proved without the Axiom of Infinity; if
not, then whether it implies Infinity; and if not, what is the
"property" that Infinity has that the other axioms don't have
that is
weaker than Infinity itself?
(Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) | ||
Theorem | tz9.1c 9245* | Alternate expression for the existence of transitive closures tz9.1 9244: the intersection of all transitive sets containing 𝐴 is a set. (Contributed by Mario Carneiro, 22-Mar-2013.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V | ||
Theorem | epfrs 9246* | The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 9247. (Contributed by Mario Carneiro, 22-Mar-2013.) |
⊢ (( E Fr 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅) | ||
Theorem | zfregs 9247* | The strong form of the Axiom of Regularity, which does not require that 𝐴 be a set. Axiom 6' of [TakeutiZaring] p. 21. See also epfrs 9246. (Contributed by NM, 17-Sep-2003.) |
⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅) | ||
Theorem | zfregs2 9248* | Alternate strong form of the Axiom of Regularity. Not every element of a nonempty class contains some element of that class. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) |
⊢ (𝐴 ≠ ∅ → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) | ||
Theorem | setind 9249* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V) | ||
Theorem | setind2 9250 | Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.) |
⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V) | ||
Syntax | ctc 9251 | Extend class notation to include the transitive closure function. |
class TC | ||
Definition | df-tc 9252* | The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)}) | ||
Theorem | tcvalg 9253* | Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9174; see tz9.1 9244.) (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}) | ||
Theorem | tcid 9254 | Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (TC‘𝐴)) | ||
Theorem | tctr 9255 | Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ Tr (TC‘𝐴) | ||
Theorem | tcmin 9256 | Defining property of the transitive closure function: it is a subset of any transitive class containing 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵)) | ||
Theorem | tc2 9257* | A variant of the definition of the transitive closure function, using instead the smallest transitive set containing 𝐴 as a member, gives almost the same set, except that 𝐴 itself must be added because it is not usually a member of (TC‘𝐴) (and it is never a member if 𝐴 is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((TC‘𝐴) ∪ {𝐴}) = ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)} | ||
Theorem | tcsni 9258 | The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (TC‘{𝐴}) = ((TC‘𝐴) ∪ {𝐴}) | ||
Theorem | tcss 9259 | The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ⊆ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴)) | ||
Theorem | tcel 9260 | The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴)) | ||
Theorem | tcidm 9261 | The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.) |
⊢ (TC‘(TC‘𝐴)) = (TC‘𝐴) | ||
Theorem | tc0 9262 | The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.) |
⊢ (TC‘∅) = ∅ | ||
Theorem | tc00 9263 | The transitive closure is empty iff its argument is. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.) |
⊢ (𝐴 ∈ 𝑉 → ((TC‘𝐴) = ∅ ↔ 𝐴 = ∅)) | ||
Syntax | cr1 9264 | Extend class definition to include the cumulative hierarchy of sets function. |
class 𝑅1 | ||
Syntax | crnk 9265 | Extend class definition to include rank function. |
class rank | ||
Definition | df-r1 9266 | Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation (𝑅1). Starting with the empty set, this function builds up layers of sets where the next layer is the power set of the previous layer (and the union of previous layers when the argument is a limit ordinal). Using the Axiom of Regularity, we can show that any set whatsoever belongs to one of the layers of this hierarchy (see tz9.13 9293). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as Theorems r10 9270, r1suc 9272, and r1lim 9274. Theorem r1val1 9288 shows a recursive definition that works for all values, and Theorems r1val2 9339 and r1val3 9340 show the value expressed in terms of rank. Other notations for this function are R with the argument as a subscript (Equation 3.1 of [BellMachover] p. 477), V with a subscript (Definition of [Enderton] p. 202), M with a subscript (Definition 15.19 of [Monk1] p. 113), the capital Greek letter psi (Definition of [Mendelson] p. 281), and bold-face R (Definition 2.1 of [Kunen] p. 95). (Contributed by NM, 2-Sep-2003.) |
⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | ||
Definition | df-rank 9267* | Define the rank function. See rankval 9318, rankval2 9320, rankval3 9342, or rankval4 9369 its value. The rank is a kind of "inverse" of the cumulative hierarchy of sets function 𝑅1: given a set, it returns an ordinal number telling us the smallest layer of the hierarchy to which the set belongs. Based on Definition 9.14 of [TakeutiZaring] p. 79. Theorem rankid 9335 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 9338. (Contributed by NM, 11-Oct-2003.) |
⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) | ||
Theorem | r1funlim 9268 | The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 9269 avoids ax-rep 5154.) (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | ||
Theorem | r1fnon 9269 | The cumulative hierarchy of sets function is a function on the class of ordinal numbers. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
⊢ 𝑅1 Fn On | ||
Theorem | r10 9270 | Value of the cumulative hierarchy of sets function at ∅. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
⊢ (𝑅1‘∅) = ∅ | ||
Theorem | r1sucg 9271 | Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | ||
Theorem | r1suc 9272 | Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | ||
Theorem | r1limg 9273* | Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥)) | ||
Theorem | r1lim 9274* | Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥)) | ||
Theorem | r1fin 9275 | The first ω levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.) |
⊢ (𝐴 ∈ ω → (𝑅1‘𝐴) ∈ Fin) | ||
Theorem | r1sdom 9276 | Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴)) | ||
Theorem | r111 9277 | The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.) |
⊢ 𝑅1:On–1-1→V | ||
Theorem | r1tr 9278 | The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ Tr (𝑅1‘𝐴) | ||
Theorem | r1tr2 9279 | The union of a cumulative hierarchy of sets at ordinal 𝐴 is a subset of the hierarchy at 𝐴. JFM CLASSES1 th. 40. (Contributed by FL, 20-Apr-2011.) |
⊢ ∪ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) | ||
Theorem | r1ordg 9280 | Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.) |
⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) | ||
Theorem | r1ord3g 9281 | Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.) |
⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) | ||
Theorem | r1ord 9282 | Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) | ||
Theorem | r1ord2 9283 | Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 22-Sep-2003.) |
⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) | ||
Theorem | r1ord3 9284 | Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) | ||
Theorem | r1sssuc 9285 | The value of the cumulative hierarchy of sets function is a subset of its value at the successor. JFM CLASSES1 Th. 39. (Contributed by FL, 20-Apr-2011.) |
⊢ (𝐴 ∈ On → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴)) | ||
Theorem | r1pwss 9286 | Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵)) | ||
Theorem | r1sscl 9287 | Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ (𝑅1‘𝐵)) | ||
Theorem | r1val1 9288* | The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)) | ||
Theorem | tz9.12lem1 9289* | Lemma for tz9.12 9292. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) ⇒ ⊢ (𝐹 “ 𝐴) ⊆ On | ||
Theorem | tz9.12lem2 9290* | Lemma for tz9.12 9292. (Contributed by NM, 22-Sep-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) ⇒ ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On | ||
Theorem | tz9.12lem3 9291* | Lemma for tz9.12 9292. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴))) | ||
Theorem | tz9.12 9292* | A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 9289 through tz9.12lem3 9291. (Contributed by NM, 22-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘𝑦)) | ||
Theorem | tz9.13 9293* | Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78. (Contributed by NM, 23-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) | ||
Theorem | tz9.13g 9294* | Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 9293 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)) | ||
Theorem | rankwflemb 9295* | Two ways of saying a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) | ||
Theorem | rankf 9296 | The domain and range of the rank function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.) |
⊢ rank:∪ (𝑅1 “ On)⟶On | ||
Theorem | rankon 9297 | The rank of a set is an ordinal number. Proposition 9.15(1) of [TakeutiZaring] p. 79. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 12-Sep-2013.) |
⊢ (rank‘𝐴) ∈ On | ||
Theorem | r1elwf 9298 | Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)) | ||
Theorem | rankvalb 9299* | Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9318 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}) | ||
Theorem | rankr1ai 9300 | One direction of rankr1a 9338. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |