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Theorem List for Metamath Proof Explorer - 40101-40200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Axiomax-frege58a 40101 If 𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2064. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
 
Theoremfrege58acor 40102 Lemma for frege59a 40103. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜓𝜒) ∧ (𝜃𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
 
Theoremfrege59a 40103 A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of [Frege1879] p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 40039 incorrectly referenced where frege30 40058 is in the original. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

(if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓𝜒) ∧ (𝜃𝜏))))
 
Theoremfrege60a 40104 Swap antecedents of ax-frege58a 40101. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
 
Theoremfrege61a 40105 Lemma for frege65a 40109. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
((if-(𝜑, 𝜓, 𝜒) → 𝜃) → ((𝜓𝜒) → 𝜃))
 
Theoremfrege62a 40106 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(if-(𝜑, 𝜓, 𝜃) → (((𝜓𝜒) ∧ (𝜃𝜏)) → if-(𝜑, 𝜒, 𝜏)))
 
Theoremfrege63a 40107 Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(if-(𝜑, 𝜓, 𝜃) → (𝜂 → (((𝜓𝜒) ∧ (𝜃𝜏)) → if-(𝜑, 𝜒, 𝜏))))
 
Theoremfrege64a 40108 Lemma for frege65a 40109. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁))))
 
Theoremfrege65a 40109 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜓𝜒) ∧ (𝜏𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
 
Theoremfrege66a 40110 Swap antecedents of frege65a 40109. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜒𝜃) ∧ (𝜂𝜁)) → (((𝜓𝜒) ∧ (𝜏𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
 
Theoremfrege67a 40111 Lemma for frege68a 40112. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
((((𝜓𝜒) ↔ 𝜃) → (𝜃 → (𝜓𝜒))) → (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))))
 
Theoremfrege68a 40112 Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))
 
20.30.3.6  _Begriffsschrift_ Chapter II with equivalence of sets
 
Theoremaxfrege52c 40113 Justification for ax-frege52c 40114. (Contributed by RP, 24-Dec-2019.)
(𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
 
Axiomax-frege52c 40114 One side of dfsbcq 3773. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
 
Theoremfrege52b 40115 The case when the content of 𝑥 is identical with the content of 𝑦 and in which a proposition controlled by an element for which we substitute the content of 𝑥 is affirmed and the same proposition, this time where we substitute the content of 𝑦, is denied does not take place. In [𝑥 / 𝑧]𝜑, 𝑥 can also occur in other than the argument (𝑧) places. Hence 𝑥 may still be contained in [𝑦 / 𝑧]𝜑. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
 
Theoremfrege53b 40116 Lemma for frege102 (via frege92 40181). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑))
 
Theoremaxfrege54c 40117 Reflexive equality of classes. Identical to eqid 2821. Justification for ax-frege54c 40118. (Contributed by RP, 24-Dec-2019.)
𝐴 = 𝐴
 
Axiomax-frege54c 40118 Reflexive equality of sets (as classes). Part of Axiom 54 of [Frege1879] p. 50. Identical to eqid 2821. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
𝐴 = 𝐴
 
Theoremfrege54b 40119 Reflexive equality of sets. The content of 𝑥 is identical with the content of 𝑥. Part of Axiom 54 of [Frege1879] p. 50. Slightly specialized version of eqid 2821. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝑥 = 𝑥
 
Theoremfrege54cor1b 40120 Reflexive equality. (Contributed by RP, 24-Dec-2019.)
[𝑥 / 𝑦]𝑦 = 𝑥
 
Theoremfrege55lem1b 40121* Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑𝑥 = 𝑧))
 
Theoremfrege55lem2b 40122 Lemma for frege55b 40123. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
 
Theoremfrege55b 40123 Lemma for frege57b 40125. Proposition 55 of [Frege1879] p. 50.

Note that eqtr2 2842 incorporates eqcom 2828 which is stronger than this proposition which is identical to equcomi 2015. Is it possible that Frege tricked himself into assuming what he was out to prove? (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

(𝑥 = 𝑦𝑦 = 𝑥)
 
Theoremfrege56b 40124 Lemma for frege57b 40125. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
 
Theoremfrege57b 40125 Analogue of frege57aid 40098. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
 
Theoremaxfrege58b 40126 If 𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2064. Justification for ax-frege58b 40127. (Contributed by RP, 28-Mar-2020.)
(∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
 
Axiomax-frege58b 40127 If 𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2064. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
(∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
 
Theoremfrege58bid 40128 If 𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2172. See ax-frege58b 40127 and frege58c 40147 for versions which more closely track the original. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)
(∀𝑥𝜑𝜑)
 
Theoremfrege58bcor 40129 Lemma for frege59b 40130. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
 
Theoremfrege59b 40130 A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of [Frege1879] p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 40039 incorrectly referenced where frege30 40058 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

([𝑦 / 𝑥]𝜑 → (¬ [𝑦 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑𝜓)))
 
Theoremfrege60b 40131 Swap antecedents of ax-frege58b 40127. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
 
Theoremfrege61b 40132 Lemma for frege65b 40136. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(([𝑥 / 𝑦]𝜑𝜓) → (∀𝑦𝜑𝜓))
 
Theoremfrege62b 40133 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 → (∀𝑥(𝜑𝜓) → [𝑦 / 𝑥]𝜓))
 
Theoremfrege63b 40134 Lemma for frege91 40180. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝑦 / 𝑥]𝜒)))
 
Theoremfrege64b 40135 Lemma for frege65b 40136. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒)))
 
Theoremfrege65b 40136 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53.

In Frege care is taken to point out that the variables in the first clauses are independent of each other and of the final term so another valid translation could be : (∀𝑥([𝑥 / 𝑎]𝜑 → [𝑥 / 𝑏]𝜓) → (∀𝑦([𝑦 / 𝑏]𝜓 → [𝑦 / 𝑐]𝜒) → ([𝑧 / 𝑎]𝜑 → [𝑧 / 𝑐]𝜒))). But that is perhaps too pedantic a translation for this exploration. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

(∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
 
Theoremfrege66b 40137 Swap antecedents of frege65b 40136. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥(𝜒𝜑) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓)))
 
Theoremfrege67b 40138 Lemma for frege68b 40139. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)))
 
Theoremfrege68b 40139 Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((∀𝑥𝜑𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))
 
20.30.3.7  _Begriffsschrift_ Chapter II with equivalence of classes

Begriffsschrift Chapter II with equivalence of classes (where they are sets).

 
Theoremfrege53c 40140 Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵[𝐵 / 𝑥]𝜑))
 
Theoremfrege54cor1c 40141* Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)
𝐴𝐶       [𝐴 / 𝑥]𝑥 = 𝐴
 
Theoremfrege55lem1c 40142* Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
((𝜑[𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑𝐴 = 𝐵))
 
Theoremfrege55lem2c 40143* Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝑥 = 𝐴[𝐴 / 𝑧]𝑧 = 𝑥)
 
Theoremfrege55c 40144 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝑥 = 𝐴𝐴 = 𝑥)
 
Theoremfrege56c 40145* Lemma for frege57c 40146. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐵𝐶       ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)))
 
Theoremfrege57c 40146* Swap order of implication in ax-frege52c 40114. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐶       (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
 
Theoremfrege58c 40147 Principle related to sp 2172. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (∀𝑥𝜑[𝐴 / 𝑥]𝜑)
 
Theoremfrege59c 40148 A kind of Aristotelian inference. Proposition 59 of [Frege1879] p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 40039 incorrectly referenced where frege30 40058 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

𝐴𝐵       ([𝐴 / 𝑥]𝜑 → (¬ [𝐴 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑𝜓)))
 
Theoremfrege60c 40149 Swap antecedents of frege58c 40147. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))
 
Theoremfrege61c 40150 Lemma for frege65c 40154. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (([𝐴 / 𝑥]𝜑𝜓) → (∀𝑥𝜑𝜓))
 
Theoremfrege62c 40151 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥]𝜓))
 
Theoremfrege63c 40152 Analogue of frege63b 40134. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)))
 
Theoremfrege64c 40153 Lemma for frege65c 40154. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (([𝐶 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓𝜒) → ([𝐶 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))
 
Theoremfrege65c 40154 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2746 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜒) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))
 
Theoremfrege66c 40155 Swap antecedents of frege65c 40154. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (∀𝑥(𝜑𝜓) → (∀𝑥(𝜒𝜑) → ([𝐴 / 𝑥]𝜒[𝐴 / 𝑥]𝜓)))
 
Theoremfrege67c 40156 Lemma for frege68c 40157. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       (((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑)))
 
Theoremfrege68c 40157 Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝐴𝐵       ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑))
 
20.30.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence

(𝑅𝐴) ⊆ 𝐴 means membership in 𝐴 is hereditary in the sequence dictated by relation 𝑅. This differs from the set-theoretic notion that a set is hereditary in a property: that all of its elements have a property and all of their elements have the property and so-on.

While the above notation is modern, it is cumbersome in the case when 𝐴 is complex and to more closely follow Frege, we abbreviate it with new notation 𝑅 hereditary 𝐴. This greatly shortens the statements for frege97 40186 and frege109 40198.

dffrege69 40158 through frege75 40164 develop this, but translation to Metamath is pending some decisions.

While Frege does not limit discussion to sets, we may have to depart from Frege by limiting 𝑅 or 𝐴 to sets when we quantify over all hereditary relations or all classes where membership is hereditary in a sequence dictated by 𝑅.

 
Theoremdffrege69 40158* If from the proposition that 𝑥 has property 𝐴 it can be inferred generally, whatever 𝑥 may be, that every result of an application of the procedure 𝑅 to 𝑥 has property 𝐴, then we say " Property 𝐴 is hereditary in the 𝑅-sequence. Definition 69 of [Frege1879] p. 55. (Contributed by RP, 28-Mar-2020.)
(∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴)
 
Theoremfrege70 40159* Lemma for frege72 40161. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑉       (𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑦(𝑋𝑅𝑦𝑦𝐴)))
 
Theoremfrege71 40160* Lemma for frege72 40161. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑉       ((∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴)) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴))))
 
Theoremfrege72 40161 If property 𝐴 is hereditary in the 𝑅-sequence, if 𝑥 has property 𝐴, and if 𝑦 is a result of an application of the procedure 𝑅 to 𝑥, then 𝑦 has property 𝐴. Proposition 72 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉       (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
 
Theoremfrege73 40162 Lemma for frege87 40176. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉       ((𝑅 hereditary 𝐴𝑋𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
 
Theoremfrege74 40163 If 𝑋 has a property 𝐴 that is hereditary in the 𝑅-sequence, then every result of a application of the procedure 𝑅 to 𝑋 has the property 𝐴. Proposition 74 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉       (𝑋𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
 
Theoremfrege75 40164* If from the proposition that 𝑥 has property 𝐴, whatever 𝑥 may be, it can be inferred that every result of an application of the procedure 𝑅 to 𝑥 has property 𝐴, then property 𝐴 is hereditary in the 𝑅-sequence. Proposition 75 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)
(∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → 𝑅 hereditary 𝐴)
 
20.30.3.9  _Begriffsschrift_ Chapter III Following in a sequence

𝑝(t+‘𝑅)𝑐 means 𝑐 follows 𝑝 in the 𝑅-sequence.

dffrege76 40165 through frege98 40187 develop this.

This will be shown to be the transitive closure of the relation 𝑅. But more work needs to be done on transitive closure of relations before this is ready for Metamath.

 
Theoremdffrege76 40165* If from the two propositions that every result of an application of the procedure 𝑅 to 𝐵 has property 𝑓 and that property 𝑓 is hereditary in the 𝑅-sequence, it can be inferred, whatever 𝑓 may be, that 𝐸 has property 𝑓, then we say 𝐸 follows 𝐵 in the 𝑅-sequence. Definition 76 of [Frege1879] p. 60.

Each of 𝐵, 𝐸 and 𝑅 must be sets. (Contributed by RP, 2-Jul-2020.)

𝐵𝑈    &   𝐸𝑉    &   𝑅𝑊       (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎𝑎𝑓) → 𝐸𝑓)) ↔ 𝐵(t+‘𝑅)𝐸)
 
Theoremfrege77 40166* If 𝑌 follows 𝑋 in the 𝑅-sequence, if property 𝐴 is hereditary in the 𝑅-sequence, and if every result of an application of the procedure 𝑅 to 𝑋 has the property 𝐴, then 𝑌 has property 𝐴. Proposition 77 of [Frege1879] p. 62. (Contributed by RP, 29-Jun-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → 𝑌𝐴)))
 
Theoremfrege78 40167* Commuted form of of frege77 40166. Proposition 78 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → (𝑋(t+‘𝑅)𝑌𝑌𝐴)))
 
Theoremfrege79 40168* Distributed form of frege78 40167. Proposition 79 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       ((𝑅 hereditary 𝐴 → ∀𝑎(𝑋𝑅𝑎𝑎𝐴)) → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌𝑌𝐴)))
 
Theoremfrege80 40169* Add additional condition to both clauses of frege79 40168. Proposition 80 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       ((𝑋𝐴 → (𝑅 hereditary 𝐴 → ∀𝑎(𝑋𝑅𝑎𝑎𝐴))) → (𝑋𝐴 → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌𝑌𝐴))))
 
Theoremfrege81 40170 If 𝑋 has a property 𝐴 that is hereditary in the 𝑅 -sequence, and if 𝑌 follows 𝑋 in the 𝑅-sequence, then 𝑌 has property 𝐴. This is a form of induction attributed to Jakob Bernoulli. Proposition 81 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (𝑋𝐴 → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌𝑌𝐴)))
 
Theoremfrege82 40171 Closed-form deduction based on frege81 40170. Proposition 82 of [Frege1879] p. 64. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       ((𝜑𝑋𝐴) → (𝑅 hereditary 𝐴 → (𝜑 → (𝑋(t+‘𝑅)𝑌𝑌𝐴))))
 
Theoremfrege83 40172 Apply commuted form of frege81 40170 when the property 𝑅 is hereditary in a disjunction of two properties, only one of which is known to be held by 𝑋. Proposition 83 of [Frege1879] p. 65. Here we introduce the union of classes where Frege has a disjunction of properties which are represented by membership in either of the classes. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑆    &   𝑌𝑇    &   𝑅𝑈    &   𝐵𝑉    &   𝐶𝑊       (𝑅 hereditary (𝐵𝐶) → (𝑋𝐵 → (𝑋(t+‘𝑅)𝑌𝑌 ∈ (𝐵𝐶))))
 
Theoremfrege84 40173 Commuted form of frege81 40170. Proposition 84 of [Frege1879] p. 65. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋(t+‘𝑅)𝑌𝑌𝐴)))
 
Theoremfrege85 40174* Commuted form of frege77 40166. Proposition 85 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (𝑋(t+‘𝑅)𝑌 → (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑅 hereditary 𝐴𝑌𝐴)))
 
Theoremfrege86 40175* Conclusion about element one past 𝑌 in the 𝑅-sequence. Proposition 86 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊    &   𝐴𝐵       (((𝑅 hereditary 𝐴𝑌𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍𝑍𝐴))) → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤𝑤𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍𝑍𝐴)))))
 
Theoremfrege87 40176* If 𝑍 is a result of an application of the procedure 𝑅 to an object 𝑌 that follows 𝑋 in the 𝑅-sequence and if every result of an application of the procedure 𝑅 to 𝑋 has a property 𝐴 that is hereditary in the 𝑅-sequence, then 𝑍 has property 𝐴. Proposition 87 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑍𝑊    &   𝑅𝑆    &   𝐴𝐵       (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤𝑤𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍𝑍𝐴))))
 
Theoremfrege88 40177* Commuted form of frege87 40176. Proposition 88 of [Frege1879] p. 67. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑍𝑊    &   𝑅𝑆    &   𝐴𝐵       (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤𝑤𝐴) → (𝑅 hereditary 𝐴𝑍𝐴))))
 
Theoremfrege89 40178* One direction of dffrege76 40165. Proposition 89 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊       (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤𝑤𝑓) → 𝑌𝑓)) → 𝑋(t+‘𝑅)𝑌)
 
Theoremfrege90 40179* Add antecedent to frege89 40178. Proposition 90 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊       ((𝜑 → ∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤𝑤𝑓) → 𝑌𝑓))) → (𝜑𝑋(t+‘𝑅)𝑌))
 
Theoremfrege91 40180 Every result of an application of a procedure 𝑅 to an object 𝑋 follows that 𝑋 in the 𝑅-sequence. Proposition 91 of [Frege1879] p. 68. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊       (𝑋𝑅𝑌𝑋(t+‘𝑅)𝑌)
 
Theoremfrege92 40181 Inference from frege91 40180. Proposition 92 of [Frege1879] p. 69. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊       (𝑋 = 𝑍 → (𝑋𝑅𝑌𝑍(t+‘𝑅)𝑌))
 
Theoremfrege93 40182* Necessary condition for two elements to be related by the transitive closure. Proposition 93 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑅𝑊       (∀𝑓(∀𝑧(𝑋𝑅𝑧𝑧𝑓) → (𝑅 hereditary 𝑓𝑌𝑓)) → 𝑋(t+‘𝑅)𝑌)
 
Theoremfrege94 40183* Looking one past a pair related by transitive closure of a relation. Proposition 94 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑍𝑉    &   𝑅𝑊       ((𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → ∀𝑓(∀𝑤(𝑋𝑅𝑤𝑤𝑓) → (𝑅 hereditary 𝑓𝑍𝑓)))) → (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌𝑋(t+‘𝑅)𝑍)))
 
Theoremfrege95 40184 Looking one past a pair related by transitive closure of a relation. Proposition 95 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑍𝑊    &   𝑅𝐴       (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌𝑋(t+‘𝑅)𝑍))
 
Theoremfrege96 40185 Every result of an application of the procedure 𝑅 to an object that follows 𝑋 in the 𝑅-sequence follows 𝑋 in the 𝑅 -sequence. Proposition 96 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑌𝑉    &   𝑍𝑊    &   𝑅𝐴       (𝑋(t+‘𝑅)𝑌 → (𝑌𝑅𝑍𝑋(t+‘𝑅)𝑍))
 
Theoremfrege97 40186 The property of following 𝑋 in the 𝑅-sequence is hereditary in the 𝑅-sequence. Proposition 97 of [Frege1879] p. 71.

Here we introduce the image of a singleton under a relation as class which stands for the property of following 𝑋 in the 𝑅 -sequence. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)

𝑋𝑈    &   𝑅𝑊       𝑅 hereditary ((t+‘𝑅) “ {𝑋})
 
Theoremfrege98 40187 If 𝑌 follows 𝑋 and 𝑍 follows 𝑌 in the 𝑅-sequence then 𝑍 follows 𝑋 in the 𝑅-sequence because the transitive closure of a relation has the transitive property. Proposition 98 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 6-Jul-2020.) (Proof modification is discouraged.)
𝑋𝐴    &   𝑌𝐵    &   𝑍𝐶    &   𝑅𝐷       (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍𝑋(t+‘𝑅)𝑍))
 
20.30.3.10  _Begriffsschrift_ Chapter III Member of sequence

𝑝((t+‘𝑅) ∪ I )𝑐 means 𝑐 is a member of the 𝑅 -sequence begining with 𝑝 and 𝑝 is a member of the 𝑅 -sequence ending with 𝑐.

dffrege99 40188 through frege114 40203 develop this.

This will be shown to be related to the transitive-reflexive closure of relation 𝑅. But more work needs to be done on transitive closure of relations before this is ready for Metamath.

 
Theoremdffrege99 40188 If 𝑍 is identical with 𝑋 or follows 𝑋 in the 𝑅 -sequence, then we say : "𝑍 belongs to the 𝑅-sequence beginning with 𝑋 " or "𝑋 belongs to the 𝑅-sequence ending with 𝑍". Definition 99 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.)
𝑍𝑈       ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
 
Theoremfrege100 40189 One direction of dffrege99 40188. Proposition 100 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑈       (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))
 
Theoremfrege101 40190 Lemma for frege102 40191. Proposition 101 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑈       ((𝑍 = 𝑋 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉))))
 
Theoremfrege102 40191 If 𝑍 belongs to the 𝑅-sequence beginning with 𝑋, then every result of an application of the procedure 𝑅 to 𝑍 follows 𝑋 in the 𝑅-sequence. Proposition 102 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝐴    &   𝑍𝐵    &   𝑉𝐶    &   𝑅𝐷       (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉𝑋(t+‘𝑅)𝑉))
 
Theoremfrege103 40192 Proposition 103 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       ((𝑍 = 𝑋𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍)))
 
Theoremfrege104 40193 Proposition 104 of [Frege1879] p. 73.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

𝑍𝑉       (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍))
 
Theoremfrege105 40194 Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)
 
Theoremfrege106 40195 Whatever follows 𝑋 in the 𝑅-sequence belongs to the 𝑅 -sequence beginning with 𝑋. Proposition 106 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝑉       (𝑋(t+‘𝑅)𝑍𝑋((t+‘𝑅) ∪ I )𝑍)
 
Theoremfrege107 40196 Proposition 107 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑉𝐴       ((𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍(t+‘𝑅)𝑉)) → (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍((t+‘𝑅) ∪ I )𝑉)))
 
Theoremfrege108 40197 If 𝑌 belongs to the 𝑅-sequence beginning with 𝑍, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑍. Proposition 108 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑍𝐴    &   𝑌𝐵    &   𝑉𝐶    &   𝑅𝐷       (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉𝑍((t+‘𝑅) ∪ I )𝑉))
 
Theoremfrege109 40198 The property of belonging to the 𝑅-sequence beginning with 𝑋 is hereditary in the 𝑅-sequence. Proposition 109 of [Frege1879] p. 74. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝑈    &   𝑅𝑉       𝑅 hereditary (((t+‘𝑅) ∪ I ) “ {𝑋})
 
Theoremfrege110 40199* Proposition 110 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
𝑋𝐴    &   𝑌𝐵    &   𝑀𝐶    &   𝑅𝐷       (∀𝑎(𝑌𝑅𝑎𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀𝑋((t+‘𝑅) ∪ I )𝑀))
 
Theoremfrege111 40200 If 𝑌 belongs to the 𝑅-sequence beginning with 𝑍, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑍 or precedes 𝑍 in the 𝑅-sequence. Proposition 111 of [Frege1879] p. 75. (Contributed by RP, 7-Jul-2020.) (Revised by RP, 8-Jul-2020.) (Proof modification is discouraged.)
𝑍𝐴    &   𝑌𝐵    &   𝑉𝐶    &   𝑅𝐷       (𝑍((t+‘𝑅) ∪ I )𝑌 → (𝑌𝑅𝑉 → (¬ 𝑉(t+‘𝑅)𝑍𝑍((t+‘𝑅) ∪ I )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 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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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