Home Metamath Proof ExplorerTheorem List (p. 83 of 425) < 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 (1-26941) Hilbert Space Explorer (26942-28466) Users' Mathboxes (28467-42420)

Theorem List for Metamath Proof Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremoiiniseg 8201 ran 𝐹 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 26-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝑁𝐴𝑀 ∈ dom 𝐹)) → ((𝐹𝑀)𝑅𝑁𝑁 ∈ ran 𝐹))

Theoremordtype2 8202 For any set-like well-ordered class, if the order isomorphism exists (is a set), then it maps some ordinal onto 𝐴 isomorphically. Otherwise, 𝐹 is a proper class, which implies that either ran 𝐹𝐴 is a proper class or dom 𝐹 = On. This weak version of ordtype 8200 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝑅 We 𝐴𝑅 Se 𝐴𝐹 ∈ V) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))

Theoremoiexg 8203 The order isomorphism on a set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       (𝐴𝑉𝐹 ∈ V)

Theoremoion 8204 The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 23-May-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       (𝐴𝑉 → dom 𝐹 ∈ On)

Theoremoiiso 8205 The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝐴𝑉𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))

Theoremoien 8206 The order type of a well-ordered set is equinumerous to the set. (Contributed by Mario Carneiro, 23-May-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝐴𝑉𝑅 We 𝐴) → dom 𝐹𝐴)

Theoremoieu 8207 Uniqueness of the unique ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝑅 We 𝐴𝑅 Se 𝐴) → ((Ord 𝐵𝐺 Isom E , 𝑅 (𝐵, 𝐴)) ↔ (𝐵 = dom 𝐹𝐺 = 𝐹)))

Theoremoismo 8208 When 𝐴 is a subclass of On, 𝐹 is a strictly monotone ordinal functions, and it is also complete (it is an isomorphism onto all of 𝐴). The proof avoids ax-rep 4597 (the second statement is trivial under ax-rep 4597). (Contributed by Mario Carneiro, 26-Jun-2015.)
𝐹 = OrdIso( E , 𝐴)       (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))

Theoremoiid 8209 The order type of an ordinal under the order is itself, and the order isomorphism is the identity function. (Contributed by Mario Carneiro, 26-Jun-2015.)
(Ord 𝐴 → OrdIso( E , 𝐴) = ( I ↾ 𝐴))

Theoremhartogslem1 8210* Lemma for hartogs 8212. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
𝐹 = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}    &   𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)}       (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥𝐴}))

Theoremhartogslem2 8211* Lemma for hartogs 8212. (Contributed by Mario Carneiro, 14-Jan-2013.)
𝐹 = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}    &   𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)}       (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)

Theoremhartogs 8212* Given any set, the Hartogs number of the set is the least ordinal not dominated by that set. This theorem proves that there is always an ordinal which satisfies this. (This theorem can be proven trivially using the AC - see theorem ondomon 9144- but this proof works in ZF.) (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)
(𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)

Theoremwofib 8213 The only sets which are well-ordered forwards and backwards are finite sets. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 23-May-2015.)
𝐴 ∈ V       ((𝑅 Or 𝐴𝐴 ∈ Fin) ↔ (𝑅 We 𝐴𝑅 We 𝐴))

Theoremwemaplem1 8214* Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}       ((𝑃𝑉𝑄𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎𝐴 ((𝑃𝑎)𝑆(𝑄𝑎) ∧ ∀𝑏𝐴 (𝑏𝑅𝑎 → (𝑃𝑏) = (𝑄𝑏)))))

Theoremwemaplem2 8215* Lemma for wemapso 8219. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑃 ∈ (𝐵𝑚 𝐴))    &   (𝜑𝑋 ∈ (𝐵𝑚 𝐴))    &   (𝜑𝑄 ∈ (𝐵𝑚 𝐴))    &   (𝜑𝑅 Or 𝐴)    &   (𝜑𝑆 Po 𝐵)    &   (𝜑𝑎𝐴)    &   (𝜑 → (𝑃𝑎)𝑆(𝑋𝑎))    &   (𝜑 → ∀𝑐𝐴 (𝑐𝑅𝑎 → (𝑃𝑐) = (𝑋𝑐)))    &   (𝜑𝑏𝐴)    &   (𝜑 → (𝑋𝑏)𝑆(𝑄𝑏))    &   (𝜑 → ∀𝑐𝐴 (𝑐𝑅𝑏 → (𝑋𝑐) = (𝑄𝑐)))       (𝜑𝑃𝑇𝑄)

Theoremwemaplem3 8216* Lemma for wemapso 8219. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑃 ∈ (𝐵𝑚 𝐴))    &   (𝜑𝑋 ∈ (𝐵𝑚 𝐴))    &   (𝜑𝑄 ∈ (𝐵𝑚 𝐴))    &   (𝜑𝑅 Or 𝐴)    &   (𝜑𝑆 Po 𝐵)    &   (𝜑𝑃𝑇𝑋)    &   (𝜑𝑋𝑇𝑄)       (𝜑𝑃𝑇𝑄)

Theoremwemappo 8217* Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values.

Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.)

𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}       ((𝐴𝑉𝑅 Or 𝐴𝑆 Po 𝐵) → 𝑇 Po (𝐵𝑚 𝐴))

Theoremwemapsolem 8218* Lemma for wemapso 8219. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}    &   𝑈 ⊆ (𝐵𝑚 𝐴)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑅 Or 𝐴)    &   (𝜑𝑆 Or 𝐵)    &   ((𝜑 ∧ ((𝑎𝑈𝑏𝑈) ∧ 𝑎𝑏)) → ∃𝑐 ∈ dom (𝑎𝑏)∀𝑑 ∈ dom (𝑎𝑏) ¬ 𝑑𝑅𝑐)       (𝜑𝑇 Or 𝑈)

Theoremwemapso 8219* Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}       ((𝐴𝑉𝑅 We 𝐴𝑆 Or 𝐵) → 𝑇 Or (𝐵𝑚 𝐴))

Theoremwemapso2lem 8220* Lemma for wemapso2 8221. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 1-Jul-2019.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}    &   𝑈 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}       (((𝐴𝑉𝑅 Or 𝐴𝑆 Or 𝐵) ∧ 𝑍𝑊) → 𝑇 Or 𝑈)

Theoremwemapso2 8221* An alternative to having a well-order on 𝑅 in wemapso 8219 is to restrict the function set to finitely-supported functions. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 1-Jul-2019.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}    &   𝑈 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}       ((𝐴𝑉𝑅 Or 𝐴𝑆 Or 𝐵) → 𝑇 Or 𝑈)

Theoremcard2on 8222* Proof that the alternate definition cardval2 8580 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)
{𝑥 ∈ On ∣ 𝑥𝐴} ∈ On

Theoremcard2inf 8223* The definition cardval2 8580 has the curious property that for non-numerable sets (for which ndmfv 6017 yields ), it still evaluates to a nonempty set, and indeed it contains ω. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
𝐴 ∈ V       (¬ ∃𝑦 ∈ On 𝑦𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥𝐴})

2.4.33  Hartogs function, order types, weak dominance

Syntaxchar 8224 Class symbol for the Hartogs/cardinal successor function.
class har

Syntaxcwdom 8225 Class symbol for the weak dominance relation.
class *

Definitiondf-har 8226* Define the Hartogs function , which maps all sets to the smallest ordinal that cannot be injected into the given set. In the important special case where 𝑥 is an ordinal, this is the cardinal successor operation.

Traditionally, the Hartogs number of a set is written ℵ(𝑋) and the cardinal successor 𝑋 +; we use functional notation for this, and cannot use the aleph symbol because it is taken for the enumerating function of the infinite initial ordinals df-aleph 8529.

Some authors define the Hartogs number of a set to be the least *infinite* ordinal which does not inject into it, thus causing the range to consist only of alephs. We use the simpler definition where the value can be any successor cardinal. (Contributed by Stefan O'Rear, 11-Feb-2015.)

har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})

Definitiondf-wdom 8227* A set is weakly dominated by a "larger" set iff the "larger" set can be mapped onto the "smaller" set or the smaller set is empty; equivalently if the smaller set can be placed into bijection with some partition of the larger set. When choice is assumed (as fodom 9107), this coincides with the 1-1 definition df-dom 7723; however, it is not known whether this is a choice-equivalent or a strictly weaker form. Some discussion of this question can be found at http://boolesrings.org/asafk/2014/on-the-partition-principle/. (Contributed by Stefan O'Rear, 11-Feb-2015.)
* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}

Theoremharf 8228 Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
har:V⟶On

Theoremharcl 8229 Closure of the Hartogs function in the ordinals. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(har‘𝑋) ∈ On

Theoremharval 8230* Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})

Theoremelharval 8231 The Hartogs number of a set is greater than all ordinals which inject into it. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
(𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌𝑋))

Theoremharndom 8232 The Hartogs number of a set does not inject into that set. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
¬ (har‘𝑋) ≼ 𝑋

Theoremharword 8233 Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.)
(𝑋𝑌 → (har‘𝑋) ⊆ (har‘𝑌))

Theoremrelwdom 8234 Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Rel ≼*

Theorembrwdom 8235* Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑌𝑉 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))

Theorembrwdomi 8236* Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
(𝑋* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))

Theorembrwdomn0 8237* Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
(𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))

Theorem0wdom 8238 Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → ∅ ≼* 𝑋)

Theoremfowdom 8239 An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
((𝐹𝑉𝐹:𝑌onto𝑋) → 𝑋* 𝑌)

Theoremwdomref 8240 Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉𝑋* 𝑋)

Theorembrwdom2 8241* Alternate characterization of the weak dominance predicate which does not require special treatment of the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑌𝑉 → (𝑋* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌𝑧 𝑧:𝑦onto𝑋))

Theoremdomwdom 8242 Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑌𝑋* 𝑌)

Theoremwdomtr 8243 Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
((𝑋* 𝑌𝑌* 𝑍) → 𝑋* 𝑍)

Theoremwdomen1 8244 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴* 𝐶𝐵* 𝐶))

Theoremwdomen2 8245 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐶* 𝐴𝐶* 𝐵))

Theoremwdompwdom 8246 Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
(𝑋* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌)

Theoremcanthwdom 8247 Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7878, equivalent to canth 6390). (Contributed by Mario Carneiro, 15-May-2015.)
¬ 𝒫 𝐴* 𝐴

Theoremwdom2d 8248* Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4597). (Contributed by Stefan O'Rear, 13-Feb-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐵 𝑥 = 𝑋)       (𝜑𝐴* 𝐵)

Theoremwdomd 8249* Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
(𝜑𝐵𝑊)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐵 𝑥 = 𝑋)       (𝜑𝐴* 𝐵)

Theorembrwdom3 8250* Condition for weak dominance with a condition reminiscent of wdomd 8249. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
((𝑋𝑉𝑌𝑊) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))

Theorembrwdom3i 8251* Weak dominance implies existence of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
(𝑋* 𝑌 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦))

Theoremunwdomg 8252 Weak dominance of a (disjoint) union. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
((𝐴* 𝐵𝐶* 𝐷 ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼* (𝐵𝐷))

Theoremxpwdomg 8253 Weak dominance of a Cartesian product. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
((𝐴* 𝐵𝐶* 𝐷) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷))

Theoremwdomima2g 8254 A set is weakly dominant over its image under any function. This version of wdomimag 8255 is stated so as to avoid ax-rep 4597. (Contributed by Mario Carneiro, 25-Jun-2015.)
((Fun 𝐹𝐴𝑉 ∧ (𝐹𝐴) ∈ 𝑊) → (𝐹𝐴) ≼* 𝐴)

Theoremwdomimag 8255 A set is weakly dominant over its image under any function. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
((Fun 𝐹𝐴𝑉) → (𝐹𝐴) ≼* 𝐴)

Theoremunxpwdom2 8256 Lemma for unxpwdom 8257. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴 × 𝐴) ≈ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))

Theoremunxpwdom 8257 If a Cartesian product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴 × 𝐴) ≼ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))

Theoremharwdom 8258 The Hartogs function is weakly dominated by 𝒫 (𝑋 × 𝑋). This follows from a more precise analysis of the bound used in hartogs 8212 to prove that (har‘𝑋) is a set. (Contributed by Mario Carneiro, 15-May-2015.)
(𝑋𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))

Theoremixpiunwdom 8259* Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 7704 this shows that 𝑥𝐴𝐵 and X𝑥𝐴𝐵 have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.)
((𝐴𝑉 𝑥𝐴 𝐵𝑊X𝑥𝐴 𝐵 ≠ ∅) → 𝑥𝐴 𝐵* (X𝑥𝐴 𝐵 × 𝐴))

2.5  ZF Set Theory - add the Axiom of Regularity

2.5.1  Introduce the Axiom of Regularity

Axiomax-reg 8260* Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 8263) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 8267). A stronger version that works for proper classes is proved as zfregs 8371. (Contributed by NM, 14-Aug-1993.)
(∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))

Theoremaxreg2 8261* Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)
(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))

Theoremzfregcl 8262* The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.)
(𝐴𝑉 → (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴))

Theoremzfreg 8263* The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form." Axiom Reg of [BellMachover] p. 480. There is also a "strong form," not requiring that 𝐴 be a set, that can be proved with more difficulty (see zfregs 8371). (Contributed by NM, 26-Nov-1995.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.)
((𝐴𝑉𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)

TheoremzfregclOLD 8264* Obsolete version of zfregcl 8262 as of 28-Apr-2021. (Contributed by NM, 5-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       (∃𝑥 𝑥𝐴 → ∃𝑥𝐴𝑦𝑥 ¬ 𝑦𝐴)

TheoremzfregOLD 8265* Obsolete version of zfreg 8263 as of 28-Apr-2021. (Contributed by NM, 26-Nov-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       (𝐴 ≠ ∅ → ∃𝑥𝐴 (𝑥𝐴) = ∅)

Theoremzfreg2OLD 8266* Alternate version of zfreg 8263 obsolete as of 28-Apr-2021. (Contributed by NM, 17-Sep-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       (𝐴 ≠ ∅ → ∃𝑥𝐴 (𝐴𝑥) = ∅)

Theoremelirrv 8267 The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr 8272 and efrirr 4913, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
¬ 𝑥𝑥

Theoremelirr 8268 No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
¬ 𝐴𝐴

Theoremsucprcreg 8269 A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.)
𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Theoremruv 8270 The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
{𝑥𝑥𝑥} = V

TheoremruALT 8271 Alternate proof of ru 3305, simplified using (indirectly) the Axiom of Regularity ax-reg 8260. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝑥𝑥} ∉ V

Theoremzfregfr 8272 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
E Fr 𝐴

Theoremen2lp 8273 No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Revised by Mario Carneiro, 25-Jun-2015.)
¬ (𝐴𝐵𝐵𝐴)

Theoremen3lplem1 8274* Lemma for en3lp 8276. (Contributed by Alan Sare, 28-Oct-2011.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Theoremen3lplem2 8275* Lemma for en3lp 8276. (Contributed by Alan Sare, 28-Oct-2011.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Theoremen3lp 8276 No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 37993 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
¬ (𝐴𝐵𝐵𝐶𝐶𝐴)

Theorempreleq 8277 Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremopthreg 8278 Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8260 (via the preleq 8277 step). See df-op 4035 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))

Theoremsuc11reg 8279 The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
(suc 𝐴 = suc 𝐵𝐴 = 𝐵)

Theoremdford2 8280* Assuming ax-reg 8260, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.)
(Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))

2.5.2  Axiom of Infinity equivalents

Theoreminf0 8281* Our Axiom of Infinity derived from existence of omega. The proof shows that the especially contrived class "ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) " exists, is a subset of its union, and contains a given set 𝑥 (and thus is nonempty). Thus, it provides an example demonstrating that a set 𝑦 exists with the necessary properties demanded by ax-inf 8298. (Contributed by NM, 15-Oct-1996.)
ω ∈ V       𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoreminf1 8282 Variation of Axiom of Infinity (using zfinf 8299 as a hypothesis). Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 14-Oct-1996.) (Revised by David Abernethy, 1-Oct-2013.)
𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))       𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))

Theoreminf2 8283* Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 8299 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)
𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))       𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)

Theoreminf3lema 8284* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ (𝐺𝐵) ↔ (𝐴𝑥 ∧ (𝐴𝑥) ⊆ 𝐵))

Theoreminf3lemb 8285* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐹‘∅) = ∅

Theoreminf3lemc 8286* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ ω → (𝐹‘suc 𝐴) = (𝐺‘(𝐹𝐴)))

Theoreminf3lemd 8287* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ ω → (𝐹𝐴) ⊆ 𝑥)

Theoreminf3lem1 8288* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 ∈ ω → (𝐹𝐴) ⊆ (𝐹‘suc 𝐴))

Theoreminf3lem2 8289* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. (Contributed by NM, 28-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐴 ∈ ω → (𝐹𝐴) ≠ 𝑥))

Theoreminf3lem3 8290* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 8263. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐴 ∈ ω → (𝐹𝐴) ≠ (𝐹‘suc 𝐴)))

Theoreminf3lem4 8291* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → (𝐴 ∈ ω → (𝐹𝐴) ⊊ (𝐹‘suc 𝐴)))

Theoreminf3lem5 8292* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ((𝐴 ∈ ω ∧ 𝐵𝐴) → (𝐹𝐵) ⊊ (𝐹𝐴)))

Theoreminf3lem6 8293* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. (Contributed by NM, 29-Oct-1996.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)

Theoreminf3lem7 8294* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8295 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 6909. (Contributed by NM, 29-Oct-1996.) (Proof shortened by Mario Carneiro, 19-Jan-2013.)
𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})    &   𝐹 = (rec(𝐺, ∅) ↾ ω)    &   𝐴 ∈ V    &   𝐵 ∈ V       ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)

Theoreminf3 8295 Our Axiom of Infinity ax-inf 8298 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 8283, and the conclusion is the version of the Axiom of Infinity shown as Axiom 7 in [TakeutiZaring] p. 43. (Other standard versions are proved later as axinf2 8300 and zfinf2 8302.) The main proof is provided by inf3lema 8284 through inf3lem7 8294, and this final piece eliminates the auxiliary hypothesis of inf3lem7 8294. This proof is due to Ian Sutherland, Richard Heck, and Norman Megill and was posted on Usenet as shown below. Although the result is not new, the authors were unable to find a published proof.
       (As posted to sci.logic on 30-Oct-1996, with annotations added.)

Theorem:  The statement "There exists a nonempty set that is a subset
of its union" implies the Axiom of Infinity.

Proof:  Let X be a nonempty set which is a subset of its union; the
latter
property is equivalent to saying that for any y in X, there exists a z
in X
such that y is in z.

Define by finite recursion a function F:omega-->(power X) such that
F_0 = 0  (See inf3lemb 8285.)
F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 8286.)
Note: ^ means intersect, < means \in ("element of").
(Finite recursion as typically done requires the existence of omega;
to avoid this we can just use transfinite recursion restricted to omega.
F is a class-term that is not necessarily a set at this point.)

Lemma 1.  F_n subset F_n+1.  (See inf3lem1 8288.)
Proof:  By induction:  F_0 subset F_1.  If y < F_n+1, then y^X subset
F_n,
so if F_n subset F_n+1, then y^X subset F_n+1, so y < F_n+2.

Lemma 2.  F_n =/= X.  (See inf3lem2 8289.)
Proof:  By induction:  F_0 =/= X because X is not empty.  Assume F_n =/=
X.
Then there is a y in X that is not in F_n.  By definition of X, there is
a
z in X that contains y.  Suppose F_n+1 = X.  Then z is in F_n+1, and z^X
contains y, so z^X is not a subset of F_n, contrary to the definition of
F_n+1.

Lemma 3.  F_n =/= F_n+1.  (See inf3lem3 8290.)
Proof:  Using the identity y^X subset F_n <-> y^(X-F_n) = 0, we have
F_n+1 = {y<X | y^(X-F_n) = 0}.  Let q = {y<X-F_n | y^(X-F_n) = 0}.
Then q subset F_n+1.  Since X-F_n is not empty by Lemma 2 and q is the
set of \in-minimal elements of X-F_n, by Foundation q is not empty, so q
and therefore F_n+1 have an element not in F_n.

Lemma 4.  F_n proper_subset F_n+1.  (See inf3lem4 8291.)
Proof:  Lemmas 1 and 3.

Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 8292.)
Proof:  Fix m and use induction on n > m.  Basis: F_m proper_subset
F_m+1
by Lemma 4.  Induction:  Assume F_m proper_subset F_n.  Then since F_n
proper_subset F_n+1, F_m proper_subset F_n+1 by transitivity of proper
subset.

By Lemma 5, F_m =/= F_n for m =/= n, so F is 1-1.  (See inf3lem6 8293.)
Thus, the inverse of F is a function with range omega and domain a
subset
of power X, so omega exists by Replacement.  (See inf3lem7 8294.)
Q.E.D.

(Contributed by NM, 29-Oct-1996.)
𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)       ω ∈ V

Theoreminfeq5i 8296 Half of infeq5 8297. (Contributed by Mario Carneiro, 16-Nov-2014.)
(ω ∈ V → ∃𝑥 𝑥 𝑥)

Theoreminfeq5 8297 The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 8303.) The left-hand side provides us with a very short way to express the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
(∃𝑥 𝑥 𝑥 ↔ ω ∈ V)

2.6  ZF Set Theory - add the Axiom of Infinity

2.6.1  Introduce the Axiom of Infinity

Axiomax-inf 8298* Axiom of Infinity. An axiom of Zermelo-Fraenkel set theory. This axiom is the gateway to "Cantor's paradise" (an expression coined by Hilbert). It asserts that given a starting set 𝑥, an infinite set 𝑦 built from it exists. Although our version is apparently not given in the literature, it is similar to, but slightly shorter than, the Axiom of Infinity in [FreydScedrov] p. 283 (see inf1 8282 and inf2 8283). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8302 and omex 8303 and are based on the (nontrivial) proof of inf3 8295. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8301. Theorem inf0 8281 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8305 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8301 requires this axiom along with Regularity ax-reg 8260 for its derivation (as theorem axinf2 8300 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8301 instead of this one. The derivation of this axiom from ax-inf2 8301 is shown by theorem axinf 8304.

Proofs should normally use the standard version ax-inf2 8301 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoremzfinf 8299* Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))

Theoremaxinf2 8300* A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 8298 and Regularity ax-reg 8260.

This theorem should not be referenced in any proof. Instead, use ax-inf2 8301 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.)

𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42420
 Copyright terms: Public domain < Previous  Next >