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Theorem List for Metamath Proof Explorer - 45001-45100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempwclaxpow 45001* Suppose 𝑀 is a transitive class that is closed under power sets intersected with 𝑀. Then, 𝑀 models the Axiom of Power Sets ax-pow 5365. One direction of Lemma II.2.8 of [Kunen2] p. 113. (Contributed by Eric Schmidt, 19-Oct-2025.)
((Tr 𝑀 ∧ ∀𝑥𝑀 (𝒫 𝑥𝑀) ∈ 𝑀) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 (∀𝑤𝑀 (𝑤𝑧𝑤𝑥) → 𝑧𝑦))
 
Theoremprclaxpr 45002* A class that is closed under the pairing operation models the Axiom of Pairing ax-pr 5432. Lemma II.2.4(4) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 29-Sep-2025.)
(∀𝑥𝑀𝑦𝑀 {𝑥, 𝑦} ∈ 𝑀 → ∀𝑥𝑀𝑦𝑀𝑧𝑀𝑤𝑀 ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧))
 
Theoremuniclaxun 45003* A class that is closed under the union operation models the Axiom of Union ax-un 7755. Lemma II.2.4(5) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 1-Oct-2025.)
(∀𝑥𝑀 𝑥𝑀 → ∀𝑥𝑀𝑦𝑀𝑧𝑀 (∃𝑤𝑀 (𝑧𝑤𝑤𝑥) → 𝑧𝑦))
 
Theoremsswfaxreg 45004* A subclass of the class of well-founded sets models the Axiom of Regularity ax-reg 9632. Lemma II.2.4(2) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 19-Oct-2025.)
(𝑀 (𝑅1 “ On) → ∀𝑥𝑀 (∃𝑦𝑀 𝑦𝑥 → ∃𝑦𝑀 (𝑦𝑥 ∧ ∀𝑧𝑀 (𝑧𝑦 → ¬ 𝑧𝑥))))
 
Theoremomssaxinf2 45005* A class that contains all ordinals up to and including ω models the Axiom of Infinity ax-inf2 9681. The antecedent of this theorem is not enough to guarantee that the class models the alternate axiom ax-inf 9678. (Contributed by Eric Schmidt, 19-Oct-2025.)
((ω ⊆ 𝑀 ∧ ω ∈ 𝑀) → ∃𝑥𝑀 (∃𝑦𝑀 (𝑦𝑥 ∧ ∀𝑧𝑀 ¬ 𝑧𝑦) ∧ ∀𝑦𝑀 (𝑦𝑥 → ∃𝑧𝑀 (𝑧𝑥 ∧ ∀𝑤𝑀 (𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
 
Theoremomelaxinf2 45006* A transitive class that contains ω models the Axiom of Infinity ax-inf2 9681. Lemma II.2.11(7) of [Kunen2] p. 114. Kunen has the additional hypotheses that the Extensionality, Separation, Pairing, and Union axioms are true in 𝑀. This, apparently, is because Kunen's statement of the Axiom of Infinity uses the defined notions and suc, and these axioms guarantee that these notions are well-defined. When we state the axiom using primitives only, the need for these hypotheses disappears.

The antecedent of this theorem is not enough to guarantee that the class models the alternate axiom ax-inf 9678. (Contributed by Eric Schmidt, 19-Oct-2025.)

((Tr 𝑀 ∧ ω ∈ 𝑀) → ∃𝑥𝑀 (∃𝑦𝑀 (𝑦𝑥 ∧ ∀𝑧𝑀 ¬ 𝑧𝑦) ∧ ∀𝑦𝑀 (𝑦𝑥 → ∃𝑧𝑀 (𝑧𝑥 ∧ ∀𝑤𝑀 (𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
 
Theoremdfac5prim 45007* dfac5 10169 expanded into primitives. (Contributed by Eric Schmidt, 19-Oct-2025.)
(CHOICE ↔ ∀𝑥((∀𝑧(𝑧𝑥 → ∃𝑤 𝑤𝑧) ∧ ∀𝑧𝑤((𝑧𝑥𝑤𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑧 → ¬ 𝑦𝑤)))) → ∃𝑦𝑧(𝑧𝑥 → ∃𝑤𝑣((𝑣𝑧𝑣𝑦) ↔ 𝑣 = 𝑤))))
 
Theoremac8prim 45008* ac8 10532 expanded into primitives. (Contributed by Eric Schmidt, 19-Oct-2025.)
((∀𝑧(𝑧𝑥 → ∃𝑤 𝑤𝑧) ∧ ∀𝑧𝑤((𝑧𝑥𝑤𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑧 → ¬ 𝑦𝑤)))) → ∃𝑦𝑧(𝑧𝑥 → ∃𝑤𝑣((𝑣𝑧𝑣𝑦) ↔ 𝑣 = 𝑤)))
 
Theoremmodelac8prim 45009* If 𝑀 is a transitive class, then the following are equivalent. (1) Every nonempty set 𝑥𝑀 of pairwise disjoint nonempty sets has a choice set in 𝑀. (2) The class 𝑀 models the Axiom of Choice, in the form ac8prim 45008.

Lemma II.2.11(7) of [Kunen2] p. 114. Kunen has the additional hypotheses that the Extensionality, Separation, Pairing, and Union axioms are true in 𝑀. This, apparently, is because Kunen's statement of the Axiom of Choice uses defined notions, including and , and these axioms guarantee that these notions are well-defined. When we state the axiom using primitives only, the need for these hypotheses disappears. (Contributed by Eric Schmidt, 19-Oct-2025.)

(Tr 𝑀 → (∀𝑥𝑀 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑀𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦)) ↔ ∀𝑥𝑀 ((∀𝑧𝑀 (𝑧𝑥 → ∃𝑤𝑀 𝑤𝑧) ∧ ∀𝑧𝑀𝑤𝑀 ((𝑧𝑥𝑤𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦𝑀 (𝑦𝑧 → ¬ 𝑦𝑤)))) → ∃𝑦𝑀𝑧𝑀 (𝑧𝑥 → ∃𝑤𝑀𝑣𝑀 ((𝑣𝑧𝑣𝑦) ↔ 𝑣 = 𝑤)))))
 
21.42.7  The class of well-founded sets is a model for ZFC
 
Theoremwfaxext 45010* The class of well-founded sets models the Axiom of Extensionality ax-ext 2708. Part of Corollary II.2.5 of [Kunen2] p. 112.

This is the first of a series of theorems showing that all the axioms of ZFC hold in the class of well-founded sets, which we here denote by 𝑊. More precisely, for each axiom of ZFC, we obtain a provable statement if we restrict all quantifiers to 𝑊 (including implicit universal quantifiers on free variables).

None of these proofs use the Axiom of Regularity. In particular, the Axiom of Regularity itself is proved to hold in 𝑊 without using Regularity. Further, the Axiom of Choice is used only in the proof that Choice holds in 𝑊. This has the consequence that any theorem of ZF (possibly proved using Regularity) can be proved, without using Regularity, to hold in 𝑊. This gives us a relative consistency result: If ZF without Regularity is consistent, so is ZF itself. Similarly, if ZFC without Regularity is consistent, so is ZFC itself. These consistency results are metatheorems and are part of Theorem II.2.13 of [Kunen2] p. 114.

(Contributed by Eric Schmidt, 11-Sep-2025.) (Revised by Eric Schmidt, 29-Sep-2025.)

𝑊 = (𝑅1 “ On)       𝑥𝑊𝑦𝑊 (∀𝑧𝑊 (𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)
 
Theoremwfaxrep 45011* The class of well-founded sets models the Axiom of Replacement ax-rep 5279. Actually, our statement is stronger, since it is an instance of Replacement only when all quantifiers in 𝑦𝜑 are relativized to 𝑊. Essentially part of Corollary II.2.5 of [Kunen2] p. 112, but note that our Replacement is different from Kunen's. (Contributed by Eric Schmidt, 29-Sep-2025.)
𝑊 = (𝑅1 “ On)       𝑥𝑊 (∀𝑤𝑊𝑦𝑊𝑧𝑊 (∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑊𝑧𝑊 (𝑧𝑦 ↔ ∃𝑤𝑊 (𝑤𝑥 ∧ ∀𝑦𝜑)))
 
Theoremwfaxsep 45012* The class of well-founded sets models the Axiom of Separation ax-sep 5296. Actually, our statement is stronger, since it is an instance of Separation only when all quantifiers in 𝜑 are relativized to 𝑊. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 29-Sep-2025.)
𝑊 = (𝑅1 “ On)       𝑧𝑊𝑦𝑊𝑥𝑊 (𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
Theoremwfaxnul 45013* The class of well-founded sets models the Null Set Axiom ax-nul 5306. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = (𝑅1 “ On)       𝑥𝑊𝑦𝑊 ¬ 𝑦𝑥
 
Theoremwfaxpow 45014* The class of well-founded sets models the Axioms of Power Sets. Part of Corollary II.2.9 of [Kunen2] p. 113. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = (𝑅1 “ On)       𝑥𝑊𝑦𝑊𝑧𝑊 (∀𝑤𝑊 (𝑤𝑧𝑤𝑥) → 𝑧𝑦)
 
Theoremwfaxpr 45015* The class of well-founded sets models the Axiom of Pairing ax-pr 5432. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 29-Sep-2025.)
𝑊 = (𝑅1 “ On)       𝑥𝑊𝑦𝑊𝑧𝑊𝑤𝑊 ((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
 
Theoremwfaxun 45016* The class of well-founded sets models the Axiom of Union ax-un 7755. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = (𝑅1 “ On)       𝑥𝑊𝑦𝑊𝑧𝑊 (∃𝑤𝑊 (𝑧𝑤𝑤𝑥) → 𝑧𝑦)
 
Theoremwfaxreg 45017* The class of well-founded sets models the Axiom of Regularity ax-reg 9632. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = (𝑅1 “ On)       𝑥𝑊 (∃𝑦𝑊 𝑦𝑥 → ∃𝑦𝑊 (𝑦𝑥 ∧ ∀𝑧𝑊 (𝑧𝑦 → ¬ 𝑧𝑥)))
 
Theoremwfaxinf2 45018* The class of well-founded sets models the Axiom of Infinity ax-inf2 9681. Part of Corollary II.2.12 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = (𝑅1 “ On)       𝑥𝑊 (∃𝑦𝑊 (𝑦𝑥 ∧ ∀𝑧𝑊 ¬ 𝑧𝑦) ∧ ∀𝑦𝑊 (𝑦𝑥 → ∃𝑧𝑊 (𝑧𝑥 ∧ ∀𝑤𝑊 (𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
 
Theoremwfac8prim 45019* The class of well-founded sets 𝑊 models the Axiom of Choice. Since the previous theorems show that all the ZF axioms hold in 𝑊, we may use any statement that ZF proves is equivalent to Choice to prove this. We use ac8prim 45008. Part of Corollary II.2.12 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 19-Oct-2025.)
𝑊 = (𝑅1 “ On)       𝑥𝑊 ((∀𝑧𝑊 (𝑧𝑥 → ∃𝑤𝑊 𝑤𝑧) ∧ ∀𝑧𝑊𝑤𝑊 ((𝑧𝑥𝑤𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦𝑊 (𝑦𝑧 → ¬ 𝑦𝑤)))) → ∃𝑦𝑊𝑧𝑊 (𝑧𝑥 → ∃𝑤𝑊𝑣𝑊 ((𝑣𝑧𝑣𝑦) ↔ 𝑣 = 𝑤)))
 
21.43  Mathbox for Glauco Siliprandi
 
21.43.1  Miscellanea
 
Theoremevth2f 45020* A version of evth2 24992 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐹    &   𝑦𝐹    &   𝑥𝑋    &   𝑦𝑋    &   𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑥) ≤ (𝐹𝑦))
 
Theoremelunif 45021* A version of eluni 4910 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐴 𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
 
Theoremrzalf 45022 A version of rzal 4509 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥 𝐴 = ∅       (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
 
Theoremfvelrnbf 45023 A version of fvelrnb 6969 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐹       (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
 
Theoremrfcnpre1 45024 If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than a given extended real B is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐵    &   𝑥𝐹    &   𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑋 = 𝐽    &   𝐴 = {𝑥𝑋𝐵 < (𝐹𝑥)}    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴𝐽)
 
Theoremubelsupr 45025* If U belongs to A and U is an upper bound, then U is the sup of A. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝐴 ⊆ ℝ ∧ 𝑈𝐴 ∧ ∀𝑥𝐴 𝑥𝑈) → 𝑈 = sup(𝐴, ℝ, < ))
 
Theoremfsumcnf 45026* A finite sum of functions to complex numbers from a common topological space is continuous, without disjoint var constraint x ph. The class expression for B normally contains free variables k and x to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ Σ𝑘𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
 
Theoremmulltgt0 45027 The product of a negative and a positive number is negative. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 · 𝐵) < 0)
 
Theoremrspcegf 45028 A version of rspcev 3622 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝜓    &   𝑥𝐴    &   𝑥𝐵    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
Theoremrabexgf 45029 A version of rabexg 5337 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 
Theoremfcnre 45030 A function continuous with respect to the standard topology, is a real mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐹𝐶)       (𝜑𝐹:𝑇⟶ℝ)
 
Theoremsumsnd 45031* A sum of a singleton is the term. The deduction version of sumsn 15782. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑘𝐵)    &   𝑘𝜑    &   ((𝜑𝑘 = 𝑀) → 𝐴 = 𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)
 
Theoremevthf 45032* A version of evth 24991 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐹    &   𝑦𝐹    &   𝑥𝑋    &   𝑦𝑋    &   𝑥𝜑    &   𝑦𝜑    &   𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
 
Theoremcnfex 45033 The class of continuous functions between two topologies is a set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
 
Theoremfnchoice 45034* For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝐴 ∈ Fin → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))
 
Theoremrefsumcn 45035* A finite sum of continuous real functions, from a common topological space, is continuous. The class expression for B normally contains free variables k and x to index it. See fsumcn 24894 for the analogous theorem on continuous complex functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ Σ𝑘𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
 
Theoremrfcnpre2 45036 If 𝐹 is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real 𝐵, is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐵    &   𝑥𝐹    &   𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑋 = 𝐽    &   𝐴 = {𝑥𝑋 ∣ (𝐹𝑥) < 𝐵}    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴𝐽)
 
Theoremcncmpmax 45037* When the hypothesis for the extreme value theorem hold, then the sup of the range of the function belongs to the range, it is real and it an upper bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑇 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑇 ≠ ∅)       (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡𝑇 (𝐹𝑡) ≤ sup(ran 𝐹, ℝ, < )))
 
Theoremrfcnpre3 45038* If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐴 = {𝑡𝑇𝐵 ≤ (𝐹𝑡)}    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴 ∈ (Clsd‘𝐽))
 
Theoremrfcnpre4 45039* If F is a continuous function with respect to the standard topology, then the preimage A of the values less than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐴 = {𝑡𝑇 ∣ (𝐹𝑡) ≤ 𝐵}    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴 ∈ (Clsd‘𝐽))
 
Theoremsumpair 45040* Sum of two distinct complex values. The class expression for 𝐴 and 𝐵 normally contain free variable 𝑘 to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑘𝐷)    &   (𝜑𝑘𝐸)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐸 ∈ ℂ)    &   (𝜑𝐴𝐵)    &   ((𝜑𝑘 = 𝐴) → 𝐶 = 𝐷)    &   ((𝜑𝑘 = 𝐵) → 𝐶 = 𝐸)       (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
 
Theoremrfcnnnub 45041* Given a real continuous function 𝐹 defined on a compact topological space, there is always a positive integer that is a strict upper bound of its range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   𝑇 = 𝐽    &   (𝜑𝑇 ≠ ∅)    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐹𝐶)       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < 𝑛)
 
Theoremrefsum2cnlem1 45042* This is the core Lemma for refsum2cn 45043: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐺    &   𝑥𝜑    &   𝐴 = (𝑘 ∈ {1, 2} ↦ if(𝑘 = 1, 𝐹, 𝐺))    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ ((𝐹𝑥) + (𝐺𝑥))) ∈ (𝐽 Cn 𝐾))
 
Theoremrefsum2cn 45043* The sum of two continuus real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐹    &   𝑥𝐺    &   𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ ((𝐹𝑥) + (𝐺𝑥))) ∈ (𝐽 Cn 𝐾))
 
Theoremadantlllr 45044 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (((((𝜑𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theorem3adantlr3 45045 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (((𝜑 ∧ (𝜓𝜒𝜂)) ∧ 𝜃) → 𝜏)
 
Theorem3adantll2 45046 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theorem3adantll3 45047 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theoremssnel 45048 If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)
 
Theoremsncldre 45049 A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → {𝐴} ∈ (Clsd‘(topGen‘ran (,))))
 
Theoremn0p 45050 A polynomial with a nonzero coefficient is not the zero polynomial. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
((𝑃 ∈ (Poly‘ℤ) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘𝑃)‘𝑁) ≠ 0) → 𝑃 ≠ 0𝑝)
 
Theorempm2.65ni 45051 Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝜑𝜓)    &   𝜑 → ¬ 𝜓)       𝜑
 
Theoremiuneq2df 45052 Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
 
Theoremnnfoctb 45053* There exists a mapping from onto any (nonempty) countable set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto𝐴)
 
Theoremelpwinss 45054 An element of the powerset of 𝐵 intersected with anything, is a subset of 𝐵. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴 ∈ (𝒫 𝐵𝐶) → 𝐴𝐵)
 
Theoremunidmex 45055 If 𝐹 is a set, then dom 𝐹 is a set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹𝑉)    &   𝑋 = dom 𝐹       (𝜑𝑋 ∈ V)
 
Theoremndisj2 45056* A non-disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑥 = 𝑦𝐵 = 𝐶)       Disj 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
 
Theoremzenom 45057 The set of integer numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ℤ ≈ ω
 
Theoremuzwo4 45058* Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑗𝜓    &   (𝑗 = 𝑘 → (𝜑𝜓))       ((𝑆 ⊆ (ℤ𝑀) ∧ ∃𝑗𝑆 𝜑) → ∃𝑗𝑆 (𝜑 ∧ ∀𝑘𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))
 
Theoremunisn0 45059 The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
{∅} = ∅
 
Theoremssin0 45060 If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) = ∅)
 
Theoreminabs3 45061 Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴𝐶))
 
Theorempwpwuni 45062 Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵))
 
Theoremdisjiun2 45063* In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑Disj 𝑥𝐴 𝐵)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷 ∈ (𝐴𝐶))    &   (𝑥 = 𝐷𝐵 = 𝐸)       (𝜑 → ( 𝑥𝐶 𝐵𝐸) = ∅)
 
Theorem0pwfi 45064 The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
∅ ∈ (𝒫 𝐴 ∩ Fin)
 
Theoremssinss2d 45065 Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐵𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)
 
Theoremzct 45066 The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ℤ ≼ ω
 
Theorempwfin0 45067 A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝒫 𝐴 ∩ Fin) ≠ ∅
 
Theoremuzct 45068 An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑍 = (ℤ𝑁)       𝑍 ≼ ω
 
Theoremiunxsnf 45069* A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐶    &   𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ {𝐴}𝐵 = 𝐶
 
Theoremfiiuncl 45070* If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐷)    &   ((𝜑𝑦𝐷𝑧𝐷) → (𝑦𝑧) ∈ 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)       (𝜑 𝑥𝐴 𝐵𝐷)
 
Theoremiunp1 45071* The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝐵    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)       (𝜑 𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = ( 𝑘 ∈ (𝑀...𝑁)𝐴𝐵))
 
Theoremfiunicl 45072* If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝜑𝑥𝐴𝑦𝐴) → (𝑥𝑦) ∈ 𝐴)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)       (𝜑 𝐴𝐴)
 
Theoremixpeq2d 45073 Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremdisjxp1 45074* The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑Disj 𝑥𝐴 𝐵)       (𝜑Disj 𝑥𝐴 (𝐵 × 𝐶))
 
Theoremdisjsnxp 45075* The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Disj 𝑗𝐴 ({𝑗} × 𝐵)
 
Theoremeliind 45076* Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 𝑥𝐵 𝐶)    &   (𝜑𝐾𝐵)    &   (𝑥 = 𝐾 → (𝐴𝐶𝐴𝐷))       (𝜑𝐴𝐷)
 
Theoremrspcef 45077 Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑥𝜓    &   𝑥𝐴    &   𝑥𝐵    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
Theoremixpssmapc 45078* An infinite Cartesian product is a subset of set exponentiation. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑥𝜑    &   (𝜑𝐶𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑X𝑥𝐴 𝐵 ⊆ (𝐶m 𝐴))
 
Theoremelintd 45079* Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)
 
Theoremssdf 45080* A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑𝐴𝐵)
 
Theorembrneqtrd 45081 Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑 → ¬ 𝐴𝑅𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑 → ¬ 𝐴𝑅𝐶)
 
Theoremssnct 45082 A set containing an uncountable set is itself uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑 → ¬ 𝐴 ≼ ω)    &   (𝜑𝐴𝐵)       (𝜑 → ¬ 𝐵 ≼ ω)
 
Theoremssuniint 45083* Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)
 
Theoremelintdv 45084* Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)
 
Theoremssd 45085* A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑𝐴𝐵)
 
Theoremralimralim 45086 Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓𝜑))
 
Theoremsnelmap 45087 Membership of the element in the range of a constant map. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → (𝐴 × {𝑥}) ∈ (𝐵m 𝐴))       (𝜑𝑥𝐵)
 
Theoremxrnmnfpnf 45088 An extended real that is neither real nor minus infinity, is plus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑 → ¬ 𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ -∞)       (𝜑𝐴 = +∞)
 
Theoremnelrnmpt 45089* Non-membership in the range of a function in maps-to notaion. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)       (𝜑 → ¬ 𝐶 ∈ ran 𝐹)
 
Theoremiuneq1i 45090 Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
𝐴 = 𝐵        𝑥𝐴 𝐶 = 𝑥𝐵 𝐶
 
Theoremnssrex 45091* Negation of subclass relationship. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)
 
Theoremssinc 45092* Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑚 ∈ (𝑀..^𝑁)) → (𝐹𝑚) ⊆ (𝐹‘(𝑚 + 1)))       (𝜑 → (𝐹𝑀) ⊆ (𝐹𝑁))
 
Theoremssdec 45093* Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑚 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑚 + 1)) ⊆ (𝐹𝑚))       (𝜑 → (𝐹𝑁) ⊆ (𝐹𝑀))
 
Theoremelixpconstg 45094* Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝐹𝑉 → (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵))
 
Theoremiineq1d 45095* Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
 
Theoremmetpsmet 45096 A metric is a pseudometric. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))
 
Theoremixpssixp 45097 Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremballss3 45098* A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   (𝜑𝐷 ∈ (PsMet‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥𝐴)       (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)
 
Theoremiunincfi 45099* Given a sequence of increasing sets, the union of a finite subsequence, is its last element. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))       (𝜑 𝑛 ∈ (𝑀...𝑁)(𝐹𝑛) = (𝐹𝑁))
 
Theoremnsstr 45100 If it's not a subclass, it's not a subclass of a smaller one. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((¬ 𝐴𝐵𝐶𝐵) → ¬ 𝐴𝐶)
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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 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49100 492 49101-49200 493 49201-49300 494 49301-49324
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