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Theorem List for Metamath Proof Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremoion 8401 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 8402 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 8403 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 8404 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 8405 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 4741 (the second statement is trivial under ax-rep 4741). (Contributed by Mario Carneiro, 26-Jun-2015.)
𝐹 = OrdIso( E , 𝐴)       (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))

Theoremoiid 8406 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 8407* Lemma for hartogs 8409. (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 8408* Lemma for hartogs 8409. (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 8409* 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 9345- 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 8410 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 8411* 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 8412* Lemma for wemapso 8416. 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 8413* Lemma for wemapso 8416. 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 8414* 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 8415* Lemma for wemapso 8416. (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 8416* 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 8417* Lemma for wemapso2 8418. (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 8418* An alternative to having a well-order on 𝑅 in wemapso 8416 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 8419* Proof that the alternate definition cardval2 8777 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)
{𝑥 ∈ On ∣ 𝑥𝐴} ∈ On

Theoremcard2inf 8420* The definition cardval2 8777 has the curious property that for non-numerable sets (for which ndmfv 6185 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 8421 Class symbol for the Hartogs/cardinal successor function.
class har

Syntaxcwdom 8422 Class symbol for the weak dominance relation.
class *

Definitiondf-har 8423* 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 8726.

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 8424* 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 9304), this coincides with the 1-1 definition df-dom 7917; 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 8425 Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
har:V⟶On

Theoremharcl 8426 Closure of the Hartogs function in the ordinals. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(har‘𝑋) ∈ On

Theoremharval 8427* Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})

Theoremelharval 8428 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 8429 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 8430 Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.)
(𝑋𝑌 → (har‘𝑋) ⊆ (har‘𝑌))

Theoremrelwdom 8431 Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Rel ≼*

Theorembrwdom 8432* Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑌𝑉 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))

Theorembrwdomi 8433* Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
(𝑋* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))

Theorembrwdomn0 8434* 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 8435 Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → ∅ ≼* 𝑋)

Theoremfowdom 8436 An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
((𝐹𝑉𝐹:𝑌onto𝑋) → 𝑋* 𝑌)

Theoremwdomref 8437 Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉𝑋* 𝑋)

Theorembrwdom2 8438* 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 8439 Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑌𝑋* 𝑌)

Theoremwdomtr 8440 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 8441 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴* 𝐶𝐵* 𝐶))

Theoremwdomen2 8442 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐶* 𝐴𝐶* 𝐵))

Theoremwdompwdom 8443 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 8444 Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 8073, equivalent to canth 6573). (Contributed by Mario Carneiro, 15-May-2015.)
¬ 𝒫 𝐴* 𝐴

Theoremwdom2d 8445* Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4741). (Contributed by Stefan O'Rear, 13-Feb-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐵 𝑥 = 𝑋)       (𝜑𝐴* 𝐵)

Theoremwdomd 8446* Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
(𝜑𝐵𝑊)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐵 𝑥 = 𝑋)       (𝜑𝐴* 𝐵)

Theorembrwdom3 8447* Condition for weak dominance with a condition reminiscent of wdomd 8446. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
((𝑋𝑉𝑌𝑊) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))

Theorembrwdom3i 8448* Weak dominance implies existence of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
(𝑋* 𝑌 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦))

Theoremunwdomg 8449 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 8450 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 8451 A set is weakly dominant over its image under any function. This version of wdomimag 8452 is stated so as to avoid ax-rep 4741. (Contributed by Mario Carneiro, 25-Jun-2015.)
((Fun 𝐹𝐴𝑉 ∧ (𝐹𝐴) ∈ 𝑊) → (𝐹𝐴) ≼* 𝐴)

Theoremwdomimag 8452 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 8453 Lemma for unxpwdom 8454. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴 × 𝐴) ≈ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))

Theoremunxpwdom 8454 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 8455 The Hartogs function is weakly dominated by 𝒫 (𝑋 × 𝑋). This follows from a more precise analysis of the bound used in hartogs 8409 to prove that (har‘𝑋) is a set. (Contributed by Mario Carneiro, 15-May-2015.)
(𝑋𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))

Theoremixpiunwdom 8456* Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 7898 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 8457* 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 8460) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 8464). A stronger version that works for proper classes is proved as zfregs 8568. (Contributed by NM, 14-Aug-1993.)
(∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))

Theoremaxreg2 8458* Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)
(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))

Theoremzfregcl 8459* 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 8460* 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 8568). (Contributed by NM, 26-Nov-1995.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.)
((𝐴𝑉𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)

TheoremzfregclOLD 8461* Obsolete version of zfregcl 8459 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 8462* Obsolete version of zfreg 8460 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 8463* Alternate version of zfreg 8460 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 8464 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 8469 and efrirr 5065, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
¬ 𝑥𝑥

Theoremelirr 8465 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 8466 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 8467 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 8468 Alternate proof of ru 3421, simplified using (indirectly) the Axiom of Regularity ax-reg 8457. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝑥𝑥} ∉ V

Theoremzfregfr 8469 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
E Fr 𝐴

Theoremen2lp 8470 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 8471* Lemma for en3lp 8473. (Contributed by Alan Sare, 28-Oct-2011.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Theoremen3lplem2 8472* Lemma for en3lp 8473. (Contributed by Alan Sare, 28-Oct-2011.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))

Theoremen3lp 8473 No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 38602 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
¬ (𝐴𝐵𝐵𝐶𝐶𝐴)

Theorempreleq 8474 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 8475 Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8457 (via the preleq 8474 step). See df-op 4162 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 8476 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 8477* Assuming ax-reg 8457, 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 8478* 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 8495. (Contributed by NM, 15-Oct-1996.)
ω ∈ V       𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoreminf1 8479 Variation of Axiom of Infinity (using zfinf 8496 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 8480* Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 8496 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 8481* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 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 8482* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 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 8483* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 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 8484* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 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 8485* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 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 8486* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 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 8487* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 for detailed description. In the proof, we invoke the Axiom of Regularity in the form of zfreg 8460. (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 8488* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 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 8489* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 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 8490* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 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 8491* Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8492 for detailed description. In the proof, we invoke the Axiom of Replacement in the form of f1dmex 7098. (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 8492 Our Axiom of Infinity ax-inf 8495 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 8480, 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 8497 and zfinf2 8499.) The main proof is provided by inf3lema 8481 through inf3lem7 8491, and this final piece eliminates the auxiliary hypothesis of inf3lem7 8491. 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 8482.)
F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 8483.)
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 8485.)
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 8486.)
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 8487.)
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 8488.)
Proof:  Lemmas 1 and 3.

Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 8489.)
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 8490.)
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 8491.)
Q.E.D.

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

Theoreminfeq5i 8493 Half of infeq5 8494. (Contributed by Mario Carneiro, 16-Nov-2014.)
(ω ∈ V → ∃𝑥 𝑥 𝑥)

Theoreminfeq5 8494 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 8500.) 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 8495* 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 8479 and inf2 8480). More standard versions, which essentially state that there exists a set containing all the natural numbers, are shown as zfinf2 8499 and omex 8500 and are based on the (nontrivial) proof of inf3 8492. This version has the advantage that when expanded to primitives, it has fewer symbols than the standard version ax-inf2 8498. Theorem inf0 8478 shows the reverse derivation of our axiom from a standard one. Theorem inf5 8502 shows a very short way to state this axiom.

The standard version of Infinity ax-inf2 8498 requires this axiom along with Regularity ax-reg 8457 for its derivation (as theorem axinf2 8497 below). In order to more easily identify the normal uses of Regularity, we will usually reference ax-inf2 8498 instead of this one. The derivation of this axiom from ax-inf2 8498 is shown by theorem axinf 8501.

Proofs should normally use the standard version ax-inf2 8498 instead of this axiom. (New usage is discouraged.) (Contributed by NM, 16-Aug-1993.)

𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoremzfinf 8496* 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 8497* A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 8495 and Regularity ax-reg 8457.

This theorem should not be referenced in any proof. Instead, use ax-inf2 8498 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)))))

Axiomax-inf2 8498* A standard version of Axiom of Infinity of ZF set theory. In English, it says: there exists a set that contains the empty set and the successors of all of its members. Theorem zfinf2 8499 shows it converted to abbreviations. This axiom was derived as theorem axinf2 8497 above, using our version of Infinity ax-inf 8495 and the Axiom of Regularity ax-reg 8457. We will reference ax-inf2 8498 instead of axinf2 8497 so that the ordinary uses of Regularity can be more easily identified. The reverse derivation of ax-inf 8495 from ax-inf2 8498 is shown by theorem axinf 8501. (Contributed by NM, 3-Nov-1996.)
𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))

Theoremzfinf2 8499* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 8498 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)
𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)

2.6.2  Existence of omega (the set of natural numbers)

Theoremomex 8500 The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. This theorem is proved assuming the Axiom of Infinity and in fact is equivalent to it, as shown by the reverse derivation inf0 8478.

A finitist (someone who doesn't believe in infinity) could, without contradiction, replace the Axiom of Infinity by its denial ¬ ω ∈ V; this would lead to ω = On by omon 7038 and Fin = V (the universe of all sets) by fineqv 8135. The finitist could still develop natural number, integer, and rational number arithmetic but would be denied the real numbers (as well as much of the rest of mathematics). In deference to the finitist, much of our development is done, when possible, without invoking the Axiom of Infinity; an example is Peano's axioms peano1 7047 through peano5 7051 (which many textbooks prove more easily assuming Infinity). (Contributed by NM, 6-Aug-1994.)

ω ∈ V

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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 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