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Theorem List for Metamath Proof Explorer - 8301-8400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoieq2 8301 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
(𝐴 = 𝐵 → OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐵))
 
Theoremnfoi 8302 Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥OrdIso(𝑅, 𝐴)
 
Theoremordiso2 8303 Generalize ordiso 8304 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)
 
Theoremordiso 8304* Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵)))
 
Theoremordtypecbv 8305* Lemma for ordtype 8320. (Contributed by Mario Carneiro, 26-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))       recs((𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = 𝐹
 
Theoremordtypelem1 8306* Lemma for ordtype 8320. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑𝑂 = (𝐹𝑇))
 
Theoremordtypelem2 8307* Lemma for ordtype 8320. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑 → Ord 𝑇)
 
Theoremordtypelem3 8308* Lemma for ordtype 8320. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑀) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
 
Theoremordtypelem4 8309* Lemma for ordtype 8320. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
 
Theoremordtypelem5 8310* Lemma for ordtype 8320. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑 → (Ord dom 𝑂𝑂:dom 𝑂𝐴))
 
Theoremordtypelem6 8311* Lemma for ordtype 8320. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       ((𝜑𝑀 ∈ dom 𝑂) → (𝑁𝑀 → (𝑂𝑁)𝑅(𝑂𝑀)))
 
Theoremordtypelem7 8312* Lemma for ordtype 8320. ran 𝑂 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → ((𝑂𝑀)𝑅𝑁𝑁 ∈ ran 𝑂))
 
Theoremordtypelem8 8313* Lemma for ordtype 8320. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
 
Theoremordtypelem9 8314* Lemma for ordtype 8320. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8318 implies that either ran 𝑂𝐴 is a proper class or dom 𝑂 = On. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑂 ∈ V)       (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
 
Theoremordtypelem10 8315* Lemma for ordtype 8320. Using ax-rep 4699, exclude the possibility that 𝑂 is a proper class and does not enumerate all of 𝐴. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
 
Theoremoi0 8316 Definition of the ordinal isomorphism when its arguments are not meaningful. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 = ∅)
 
Theoremoicl 8317 The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       Ord dom 𝐹
 
Theoremoif 8318 The order isomorphism of the well-order 𝑅 on 𝐴 is a function. (Contributed by Mario Carneiro, 23-May-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       𝐹:dom 𝐹𝐴
 
Theoremoiiso2 8319 The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism onto ran 𝑂 (which is a subset of 𝐴 by oif 8318). (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, ran 𝐹))
 
Theoremordtype 8320 For any set-like well-ordered class, there is an isomorphic ordinal number called its order type. (Contributed by Jeff Hankins, 17-Oct-2009.) (Revised by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
 
Theoremoiiniseg 8321 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 8322 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 8320 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 8323 The order isomorphism on a set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       (𝐴𝑉𝐹 ∈ V)
 
Theoremoion 8324 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 8325 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 8326 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 8327 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 8328 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 4699 (the second statement is trivial under ax-rep 4699). (Contributed by Mario Carneiro, 26-Jun-2015.)
𝐹 = OrdIso( E , 𝐴)       (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))
 
Theoremoiid 8329 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 8330* Lemma for hartogs 8332. (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 8331* Lemma for hartogs 8332. (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 8332* 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 9264- 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 8333 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 8334* 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 8335* Lemma for wemapso 8339. 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 8336* Lemma for wemapso 8339. 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 8337* 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 8338* Lemma for wemapso 8339. (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 8339* 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 8340* Lemma for wemapso2 8341. (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 8341* An alternative to having a well-order on 𝑅 in wemapso 8339 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 8342* Proof that the alternate definition cardval2 8700 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)
{𝑥 ∈ On ∣ 𝑥𝐴} ∈ On
 
Theoremcard2inf 8343* The definition cardval2 8700 has the curious property that for non-numerable sets (for which ndmfv 6128 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 8344 Class symbol for the Hartogs/cardinal successor function.
class har
 
Syntaxcwdom 8345 Class symbol for the weak dominance relation.
class *
 
Definitiondf-har 8346* 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 8649.

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 8347* 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 9227), this coincides with the 1-1 definition df-dom 7843; 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 8348 Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
har:V⟶On
 
Theoremharcl 8349 Closure of the Hartogs function in the ordinals. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(har‘𝑋) ∈ On
 
Theoremharval 8350* Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
 
Theoremelharval 8351 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 8352 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 8353 Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.)
(𝑋𝑌 → (har‘𝑋) ⊆ (har‘𝑌))
 
Theoremrelwdom 8354 Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Rel ≼*
 
Theorembrwdom 8355* Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑌𝑉 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
 
Theorembrwdomi 8356* Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
(𝑋* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))
 
Theorembrwdomn0 8357* 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 8358 Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉 → ∅ ≼* 𝑋)
 
Theoremfowdom 8359 An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
((𝐹𝑉𝐹:𝑌onto𝑋) → 𝑋* 𝑌)
 
Theoremwdomref 8360 Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑉𝑋* 𝑋)
 
Theorembrwdom2 8361* 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 8362 Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
(𝑋𝑌𝑋* 𝑌)
 
Theoremwdomtr 8363 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 8364 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴* 𝐶𝐵* 𝐶))
 
Theoremwdomen2 8365 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐶* 𝐴𝐶* 𝐵))
 
Theoremwdompwdom 8366 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 8367 Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7998, equivalent to canth 6508). (Contributed by Mario Carneiro, 15-May-2015.)
¬ 𝒫 𝐴* 𝐴
 
Theoremwdom2d 8368* Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4699). (Contributed by Stefan O'Rear, 13-Feb-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐵 𝑥 = 𝑋)       (𝜑𝐴* 𝐵)
 
Theoremwdomd 8369* Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
(𝜑𝐵𝑊)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐵 𝑥 = 𝑋)       (𝜑𝐴* 𝐵)
 
Theorembrwdom3 8370* Condition for weak dominance with a condition reminiscent of wdomd 8369. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
((𝑋𝑉𝑌𝑊) → (𝑋* 𝑌 ↔ ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦)))
 
Theorembrwdom3i 8371* Weak dominance implies existence of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
(𝑋* 𝑌 → ∃𝑓𝑥𝑋𝑦𝑌 𝑥 = (𝑓𝑦))
 
Theoremunwdomg 8372 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 8373 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 8374 A set is weakly dominant over its image under any function. This version of wdomimag 8375 is stated so as to avoid ax-rep 4699. (Contributed by Mario Carneiro, 25-Jun-2015.)
((Fun 𝐹𝐴𝑉 ∧ (𝐹𝐴) ∈ 𝑊) → (𝐹𝐴) ≼* 𝐴)
 
Theoremwdomimag 8375 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 8376 Lemma for unxpwdom 8377. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴 × 𝐴) ≈ (𝐵𝐶) → (𝐴* 𝐵𝐴𝐶))
 
Theoremunxpwdom 8377 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 8378 The Hartogs function is weakly dominated by 𝒫 (𝑋 × 𝑋). This follows from a more precise analysis of the bound used in hartogs 8332 to prove that (har‘𝑋) is a set. (Contributed by Mario Carneiro, 15-May-2015.)
(𝑋𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋))
 
Theoremixpiunwdom 8379* Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 7824 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 8380* 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 8383) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 8387). A stronger version that works for proper classes is proved as zfregs 8491. (Contributed by NM, 14-Aug-1993.)
(∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))
 
Theoremaxreg2 8381* Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.)
(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))
 
Theoremzfregcl 8382* 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 8383* 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 8491). (Contributed by NM, 26-Nov-1995.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.)
((𝐴𝑉𝐴 ≠ ∅) → ∃𝑥𝐴 (𝑥𝐴) = ∅)
 
TheoremzfregclOLD 8384* Obsolete version of zfregcl 8382 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 8385* Obsolete version of zfreg 8383 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 8386* Alternate version of zfreg 8383 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 8387 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 8392 and efrirr 5019, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
¬ 𝑥𝑥
 
Theoremelirr 8388 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 8389 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 8390 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 8391 Alternate proof of ru 3401, simplified using (indirectly) the Axiom of Regularity ax-reg 8380. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝑥𝑥} ∉ V
 
Theoremzfregfr 8392 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
E Fr 𝐴
 
Theoremen2lp 8393 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 8394* Lemma for en3lp 8396. (Contributed by Alan Sare, 28-Oct-2011.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
 
Theoremen3lplem2 8395* Lemma for en3lp 8396. (Contributed by Alan Sare, 28-Oct-2011.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅))
 
Theoremen3lp 8396 No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 38102 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
¬ (𝐴𝐵𝐵𝐶𝐶𝐴)
 
Theorempreleq 8397 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 8398 Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8380 (via the preleq 8397 step). See df-op 4132 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 8399 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 8400* Assuming ax-reg 8380, 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)))
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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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