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Theorem List for Metamath Proof Explorer - 40301-40400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcotrclrtrcl 40301 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
(t+ ∘ t*) = t*

Theoremcortrclrtrcl 40302 The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(t* ∘ t*) = t*

20.31.2.5  Adapted from Frege

Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879].

Theoremfrege77d 40303 If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 77 of [Frege1879] p. 62. Compare with frege77 40498. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)    &   (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈)       (𝜑𝐵𝑈)

Theoremfrege81d 40304 If the image of 𝑈 is a subset 𝑈, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 81 of [Frege1879] p. 63. Compare with frege81 40502. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)       (𝜑𝐵𝑈)

Theoremfrege83d 40305 If the image of the union of 𝑈 and 𝑉 is a subset of the union of 𝑈 and 𝑉, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of the union of 𝑈 and 𝑉. Similar to Proposition 83 of [Frege1879] p. 65. Compare with frege83 40504. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅 “ (𝑈𝑉)) ⊆ (𝑈𝑉))       (𝜑𝐵 ∈ (𝑈𝑉))

Theoremfrege96d 40306 If 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 40517. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)

Theoremfrege87d 40307 If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 87 of [Frege1879] p. 66. Compare with frege87 40508. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶𝑅𝐵)    &   (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)       (𝜑𝐵𝑈)

Theoremfrege91d 40308 If 𝐵 follows 𝐴 in 𝑅 then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 91 of [Frege1879] p. 68. Comparw with frege91 40512. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)

Theoremfrege97d 40309 If 𝐴 contains all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 97 of [Frege1879] p. 71. Compare with frege97 40518. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 = ((t+‘𝑅) “ 𝑈))       (𝜑 → (𝑅𝐴) ⊆ 𝐴)

Theoremfrege98d 40310 If 𝐶 follows 𝐴 and 𝐵 follows 𝐶 in the transitive closure of 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 40519. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶(t+‘𝑅)𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)

Theoremfrege102d 40311 If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 40523. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)

Theoremfrege106d 40312 If 𝐵 follows 𝐴 in 𝑅, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in 𝑅. Similar to Proposition 106 of [Frege1879] p. 73. Compare with frege106 40527. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴𝑅𝐵)       (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵))

Theoremfrege108d 40313 If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 108 of [Frege1879] p. 74. Compare with frege108 40529. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))    &   (𝜑𝐶𝑅𝐵)       (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))

Theoremfrege109d 40314 If 𝐴 contains all elements of 𝑈 and all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 109 of [Frege1879] p. 74. Compare with frege109 40530. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))       (𝜑 → (𝑅𝐴) ⊆ 𝐴)

Theoremfrege114d 40315 If either 𝑅 relates 𝐴 and 𝐵 or 𝐴 and 𝐵 are the same, then either 𝐴 and 𝐵 are the same, 𝑅 relates 𝐴 and 𝐵, 𝑅 relates 𝐵 and 𝐴. Similar to Proposition 114 of [Frege1879] p. 76. Compare with frege114 40535. (Contributed by RP, 15-Jul-2020.)
(𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵))       (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵𝐵𝑅𝐴))

Theoremfrege111d 40316 If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐴 follows 𝐵 or 𝐵 and 𝐴 in the transitive closure of 𝑅. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 40532. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))    &   (𝜑𝐶𝑅𝐵)       (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵𝐵(t+‘𝑅)𝐴))

Theoremfrege122d 40317 If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 is the successor of 𝑋, then 𝐴 and 𝐵 are the same (or 𝐵 follows 𝐴 in the transitive closure of 𝐹). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 40543. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴 = (𝐹𝑋))    &   (𝜑𝐵 = (𝐹𝑋))       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))

Theoremfrege124d 40318 If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 124 of [Frege1879] p. 80. Compare with frege124 40545. (Contributed by RP, 16-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝑋 ∈ dom 𝐹)    &   (𝜑𝐴 = (𝐹𝑋))    &   (𝜑𝑋(t+‘𝐹)𝐵)    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))

Theoremfrege126d 40319 If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 126 of [Frege1879] p. 81. Compare with frege126 40547. (Contributed by RP, 16-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝑋 ∈ dom 𝐹)    &   (𝜑𝐴 = (𝐹𝑋))    &   (𝜑𝑋(t+‘𝐹)𝐵)    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵𝐵(t+‘𝐹)𝐴))

Theoremfrege129d 40320 If 𝐹 is a function and (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹, the successor of 𝐴 is either 𝐵 or it follows 𝐵 or it comes before 𝐵 in the transitive closure of 𝐹. Similar to Proposition 129 of [Frege1879] p. 83. Comparw with frege129 40550. (Contributed by RP, 16-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝐴 ∈ dom 𝐹)    &   (𝜑𝐶 = (𝐹𝐴))    &   (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵𝐵(t+‘𝐹)𝐴))    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐵(t+‘𝐹)𝐶𝐵 = 𝐶𝐶(t+‘𝐹)𝐵))

Theoremfrege131d 40321 If 𝐹 is a function and 𝐴 contains all elements of 𝑈 and all elements before or after those elements of 𝑈 in the transitive closure of 𝐹, then the image under 𝐹 of 𝐴 is a subclass of 𝐴. Similar to Proposition 131 of [Frege1879] p. 85. Compare with frege131 40552. (Contributed by RP, 17-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝐴 = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐹𝐴) ⊆ 𝐴)

Theoremfrege133d 40322 If 𝐹 is a function and 𝐴 and 𝐵 both follow 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹 (or both if it loops). Similar to Proposition 133 of [Frege1879] p. 86. Compare with frege133 40554. (Contributed by RP, 18-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝑋(t+‘𝐹)𝐴)    &   (𝜑𝑋(t+‘𝐹)𝐵)    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵𝐵(t+‘𝐹)𝐴))

20.31.3  Propositions from _Begriffsschrift_

In 1879, Frege introduced notation for documenting formal reasoning about propositions (and classes) which covered elements of propositional logic, predicate calculus and reasoning about relations. However, due to the pitfalls of naive set theory, adapting this work for inclusion in set.mm required dividing statements about propositions from those about classes and identifying when a restriction to sets is required. For an overview comparing the details of Frege's two-dimensional notation and that used in set.mm, see mmfrege.html. See ru 3757 for discussion of an example of a class that is not a set.

Numbered propositions from [Frege1879]. ax-frege1 40348, ax-frege2 40349, ax-frege8 40367, ax-frege28 40388, ax-frege31 40392, ax-frege41 40403, frege52 (see ax-frege52a 40415, frege52b 40447, and ax-frege52c 40446 for translations), frege54 (see ax-frege54a 40420, frege54b 40451 and ax-frege54c 40450 for translations) and frege58 (see ax-frege58a 40433, ax-frege58b 40459 and frege58c 40479 for translations) are considered "core" or axioms. However, at least ax-frege8 40367 can be derived from ax-frege1 40348 and ax-frege2 40349, see axfrege8 40365.

Frege introduced implication, negation and the universal quantifier as primitives and did not in the numbered propositions use other logical connectives other than equivalence introduced in ax-frege52a 40415, frege52b 40447, and ax-frege52c 40446. In dffrege69 40490, Frege introduced 𝑅 hereditary 𝐴 to say that relation 𝑅, when restricted to operate on elements of class 𝐴, will only have elements of class 𝐴 in its domain; see df-he 40331 for a definition in terms of image and subset. In dffrege76 40497, Frege introduced notation for the concept of two sets related by the transitive closure of a relation, for which we write 𝑋(t+‘𝑅)𝑌, which requires 𝑅 to also be a set. In dffrege99 40520, Frege introduced notation for the concept of two sets either identical or related by the transitive closure of a relation, for which we write 𝑋((t+‘𝑅) ∪ I )𝑌, which is a superclass of sets related by the reflexive-transitive relation 𝑋(t*‘𝑅)𝑌. Finally, in dffrege115 40536, Frege introduced notation for the concept of a relation having the property elements in its domain pair up with only one element each in its range, for which we write Fun 𝑅 (to ignore any non-relational content of the class 𝑅). Frege did this without the expressing concept of a relation (or its transitive closure) as a class, and needed to invent conventions for discussing indeterminate propositions with two slots free and how to recognize which of the slots was domain and which was range. See mmfrege.html 40536 for details.

English translations for specific propositions lifted in part from a translation by Stefan Bauer-Mengelberg as reprinted in From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. An attempt to align these propositions in the larger set.mm database has also been made. See frege77d 40303 for an example.

20.31.3.1  _Begriffsschrift_ Chapter I

Section 2 introduces the turnstile which turns an idea which may be true 𝜑 into an assertion that it does hold true 𝜑. Section 5 introduces implication, (𝜑𝜓). Section 6 introduces the single rule of interference relied upon, modus ponens ax-mp 5. Section 7 introduces negation and with in synonyms for or 𝜑𝜓), and ¬ (𝜑 → ¬ 𝜓), and two for exclusive-or corresponding to df-or 845, df-an 400, dfxor4 40323, dfxor5 40324.

Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biimplication (𝜑𝜓) or class equality 𝐴 = 𝐵 in this adaptation. Section 10 introduces "truth functions" for one or two variables in equally troubling notation, as the arguments may be understood to be logical predicates or collections. Here f(𝜑) is interpreted to mean if-(𝜑, 𝜓, 𝜒) where the content of the "function" is specified by the latter two argments or logical equivalent, while g(𝐴) is read as 𝐴𝐺 and h(𝐴, 𝐵) as 𝐴𝐻𝐵. This necessarily introduces a need for set theory as both 𝐴𝐺 and 𝐴𝐻𝐵 cannot hold unless 𝐴 is a set. (Also 𝐵.)

Section 11 introduces notation for generality, but there is no standard notation for generality when the variable is a proposition because it was realized after Frege that the universe of all possible propositions includes paradoxical constructions leading to the failure of naive set theory. So adopting f(𝜑) as if-(𝜑, 𝜓, 𝜒) would result in the translation of 𝜑 f (𝜑) as (𝜓𝜒). For collections, we must generalize over set variables or run into the same problems; this leads to 𝐴 g(𝐴) being translated as 𝑎𝑎𝐺 and so forth.

Under this interpreation the text of section 11 gives us sp 2184 (or simpl 486 and simpr 488 and anifp 1067 in the propositional case) and statements similar to cbvalivw 2015, ax-gen 1797, alrimiv 1929, and alrimdv 1931. These last four introduce a generality and have no useful definition in terms of propositional variables.

Section 12 introduces some combinations of primitive symbols and their human language counterparts. Using class notation, these can also be expressed without dummy variables. All are A, 𝑥𝑥𝐴, ¬ ∃𝑥¬ 𝑥𝐴 alex 1827, 𝐴 = V eqv 3488; Some are not B, ¬ ∀𝑥𝑥𝐵, 𝑥¬ 𝑥𝐵 exnal 1828, 𝐵 ⊊ V pssv 4381, 𝐵 ≠ V nev 40327; There are no C, 𝑥¬ 𝑥𝐶, ¬ ∃𝑥𝑥𝐶 alnex 1783, 𝐶 = ∅ eq0 4291; There exist D, ¬ ∀𝑥¬ 𝑥𝐷, 𝑥𝑥𝐷 df-ex 1782, ∅ ⊊ 𝐷 0pss 4379, 𝐷 ≠ ∅ n0 4293.

Notation for relations between expressions also can be written in various ways. All E are P, 𝑥(𝑥𝐸𝑥𝑃), ¬ ∃𝑥(𝑥𝐸 ∧ ¬ 𝑥𝑃) dfss6 3942, 𝐸 = (𝐸𝑃) df-ss 3936, 𝐸𝑃 dfss2 3939; No F are P, 𝑥(𝑥𝐹 → ¬ 𝑥𝑃), ¬ ∃𝑥(𝑥𝐹𝑥𝑃) alinexa 1844, (𝐹𝑃) = ∅ disj1 4384; Some G are not P, ¬ ∀𝑥(𝑥𝐺𝑥𝑃), 𝑥(𝑥𝐺 ∧ ¬ 𝑥𝑃) exanali 1860, (𝐺𝑃) ⊊ 𝐺 nssinpss 4218, ¬ 𝐺𝑃 nss 4015; Some H are P, ¬ ∀𝑥(𝑥𝐻 → ¬ 𝑥𝑃), 𝑥(𝑥𝐻𝑥𝑃) exnalimn 1845, ∅ ⊊ (𝐻𝑃) 0pssin 40328, (𝐻𝑃) ≠ ∅ ndisj 4310.

Theoremdfxor4 40323 Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) ↔ ¬ ((¬ 𝜑𝜓) → ¬ (𝜑 → ¬ 𝜓)))

Theoremdfxor5 40324 Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) ↔ ¬ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑𝜓)))

Theoremdf3or2 40325 Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.)
((𝜑𝜓𝜒) ↔ (¬ 𝜑 → (¬ 𝜓𝜒)))

Theoremdf3an2 40326 Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.)
((𝜑𝜓𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))

Theoremnev 40327* Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
(𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥𝐴)

Theorem0pssin 40328* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
(∅ ⊊ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))

20.31.3.2  _Begriffsschrift_ Notation hints

The statement 𝑅 hereditary 𝐴 means relation 𝑅 is hereditary (in the sense of Frege) in the class 𝐴 or (𝑅𝐴) ⊆ 𝐴. The former is only a slight reduction in the number of symbols, but this reduces the number of floating hypotheses needed to be checked.

As Frege was not using the language of classes or sets, this naturally 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.

Theoremrp-imass 40329 If the 𝑅-image of a class 𝐴 is a subclass of 𝐵, then the restriction of 𝑅 to 𝐴 is a subset of the Cartesian product of 𝐴 and 𝐵. (Contributed by RP, 24-Dec-2019.)
((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))

Syntaxwhe 40330 The property of relation 𝑅 being hereditary in class 𝐴.
wff 𝑅 hereditary 𝐴

Definitiondf-he 40331 The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
(𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)

Theoremdfhe2 40332 The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
(𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐴))

Theoremdfhe3 40333* The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
(𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))

Theoremheeq12 40334 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
((𝑅 = 𝑆𝐴 = 𝐵) → (𝑅 hereditary 𝐴𝑆 hereditary 𝐵))

Theoremheeq1 40335 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
(𝑅 = 𝑆 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))

Theoremheeq2 40336 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
(𝐴 = 𝐵 → (𝑅 hereditary 𝐴𝑅 hereditary 𝐵))

Theoremsbcheg 40337 Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶))

Theoremhess 40338 Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
(𝑆𝑅 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))

Theoremxphe 40339 Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) (Proof shortened by OpenAI, 3-Jul-2020.)
(𝐴 × 𝐵) hereditary 𝐵

Theorem0he 40340 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
∅ hereditary 𝐴

Theorem0heALT 40341 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
∅ hereditary 𝐴

Theoremhe0 40342 Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
𝐴 hereditary ∅

Theoremunhe1 40343 The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.)
((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → (𝑅𝑆) hereditary 𝐴)

Theoremsnhesn 40344 Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
{⟨𝐴, 𝐴⟩} hereditary {𝐵}

Theoremidhe 40345 The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
I hereditary 𝐴

Theorempsshepw 40346 The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
[] hereditary 𝒫 𝐴

Theoremsshepw 40347 The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
( [] ∪ I ) hereditary 𝒫 𝐴

20.31.3.3  _Begriffsschrift_ Chapter II Implication

Axiomax-frege1 40348 The case in which 𝜑 is denied, 𝜓 is affirmed, and 𝜑 is affirmed is excluded. This is evident since 𝜑 cannot at the same time be denied and affirmed. Axiom 1 of [Frege1879] p. 26. Identical to ax-1 6. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))

Axiomax-frege2 40349 If a proposition 𝜒 is a necessary consequence of two propositions 𝜓 and 𝜑 and one of those, 𝜓, is in turn a necessary consequence of the other, 𝜑, then the proposition 𝜒 is a necessary consequence of the latter one, 𝜑, alone. Axiom 2 of [Frege1879] p. 26. Identical to ax-2 7. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))

Theoremrp-simp2-frege 40350 Simplification of triple conjunction. Compare with simp2 1134. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜓)))

Theoremrp-simp2 40351 Simplification of triple conjunction. Identical to simp2 1134. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓𝜒) → 𝜓)

Theoremrp-frege3g 40352 Add antecedent to ax-frege2 40349. More general statement than frege3 40353. Like ax-frege2 40349, it is essentially a closed form of mpd 15, however it has an extra antecedent.

It would be more natural to prove from a1i 11 and ax-frege2 40349 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

(𝜑 → ((𝜓 → (𝜒𝜃)) → ((𝜓𝜒) → (𝜓𝜃))))

Theoremfrege3 40353 Add antecedent to ax-frege2 40349. Special case of rp-frege3g 40352. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒 → (𝜑𝜓)) → ((𝜒𝜑) → (𝜒𝜓))))

Theoremrp-misc1-frege 40354 Double-use of ax-frege2 40349. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑 → (𝜓𝜒)) → (𝜑𝜓)) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))

Theoremrp-frege24 40355 Introducing an embedded antecedent. Alternate proof for frege24 40373. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))

Theoremrp-frege4g 40356 Deduction related to distribution. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜑 → ((𝜓𝜒) → (𝜓𝜃))))

Theoremfrege4 40357 Special case of closed form of a2d 29. Special case of rp-frege4g 40356. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (𝜒 → (𝜑𝜓))) → ((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓))))

Theoremfrege5 40358 A closed form of syl 17. Identical to imim2 58. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theoremrp-7frege 40359 Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓𝜒)) → (𝜃 → ((𝜑𝜓) → (𝜑𝜒))))

Theoremrp-4frege 40360 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
((𝜑 → ((𝜓𝜑) → 𝜒)) → (𝜑𝜒))

Theoremrp-6frege 40361 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
(𝜑 → ((𝜓 → ((𝜒𝜓) → 𝜃)) → (𝜓𝜃)))

Theoremrp-8frege 40362 Eliminate antecedent when it is implied by previous antecedent. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓 → ((𝜒𝜓) → 𝜃))) → (𝜑 → (𝜓𝜃)))

Theoremrp-frege25 40363 Closed form for a1dd 50. Alternate route to Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))

Theoremfrege6 40364 A closed form of imim2d 57 which is a deduction adding nested antecedents. Proposition 6 of [Frege1879] p. 33. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → ((𝜃𝜓) → (𝜃𝜒))))

Theoremaxfrege8 40365 Swap antecedents. Identical to pm2.04 90. This demonstrates that Axiom 8 of [Frege1879] p. 35 is redundant.

Proof follows closely proof of pm2.04 90 in https://us.metamath.org/mmsolitaire/pmproofs.txt 90, but in the style of Frege's 1879 work. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Theoremfrege7 40366 A closed form of syl6 35. The first antecedent is used to replace the consequent of the second antecedent. Proposition 7 of [Frege1879] p. 34. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒 → (𝜃𝜑)) → (𝜒 → (𝜃𝜓))))

Axiomax-frege8 40367 Swap antecedents. If two conditions have a proposition as a consequence, their order is immaterial. Third axiom of Frege's 1879 work but identical to pm2.04 90 which can be proved from only ax-mp 5, ax-frege1 40348, and ax-frege2 40349. (Redundant) Axiom 8 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Theoremfrege26 40368 Identical to idd 24. Proposition 26 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜓))

Theoremfrege27 40369 We cannot (at the same time) affirm 𝜑 and deny 𝜑. Identical to id 22. Proposition 27 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑𝜑)

Theoremfrege9 40370 Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 40358 only in an unessential way. Identical to imim1 83. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))

Theoremfrege12 40371 A closed form of com23 86. Proposition 12 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜑 → (𝜒 → (𝜓𝜃))))

Theoremfrege11 40372 Elimination of a nested antecedent as a partial converse of ja 189. If the proposition that 𝜓 takes place or 𝜑 does not is a sufficient condition for 𝜒, then 𝜓 by itself is a sufficient condition for 𝜒. Identical to jarr 106. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))

Theoremfrege24 40373 Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 40355 which was proved without relying on ax-frege8 40367. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))

Theoremfrege16 40374 A closed form of com34 91. Proposition 16 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃𝜏)))) → (𝜑 → (𝜓 → (𝜃 → (𝜒𝜏)))))

Theoremfrege25 40375 Closed form for a1dd 50. Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))

Theoremfrege18 40376 Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜃𝜑) → (𝜓 → (𝜃𝜒))))

Theoremfrege22 40377 A closed form of com45 97. Proposition 22 of [Frege1879] p. 41. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))) → (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜃𝜂))))))

Theoremfrege10 40378 Result commuting antecedents within an antecedent. Proposition 10 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑 → (𝜓𝜒)) → 𝜃) → ((𝜓 → (𝜑𝜒)) → 𝜃))

Theoremfrege17 40379 A closed form of com3l 89. Proposition 17 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜓 → (𝜒 → (𝜑𝜃))))

Theoremfrege13 40380 A closed form of com3r 87. Proposition 13 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜒 → (𝜑 → (𝜓𝜃))))

Theoremfrege14 40381 Closed form of a deduction based on com3r 87. Proposition 14 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃𝜏)))) → (𝜑 → (𝜃 → (𝜓 → (𝜒𝜏)))))

Theoremfrege19 40382 A closed form of syl6 35. Proposition 19 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜒𝜃) → (𝜑 → (𝜓𝜃))))

Theoremfrege23 40383 Syllogism followed by rotation of three antecedents. Proposition 23 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → ((𝜏𝜑) → (𝜓 → (𝜒 → (𝜏𝜃)))))

Theoremfrege15 40384 A closed form of com4r 94. Proposition 15 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃𝜏)))) → (𝜃 → (𝜑 → (𝜓 → (𝜒𝜏)))))

Theoremfrege21 40385 Replace antecedent in antecedent. Proposition 21 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → 𝜒) → ((𝜑𝜃) → ((𝜃𝜓) → 𝜒)))

Theoremfrege20 40386 A closed form of syl8 76. Proposition 20 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → ((𝜃𝜏) → (𝜑 → (𝜓 → (𝜒𝜏)))))

20.31.3.4  _Begriffsschrift_ Chapter II Implication and Negation

Theoremaxfrege28 40387 Contraposition. Identical to con3 156. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Axiomax-frege28 40388 Contraposition. Identical to con3 156. Theorem *2.16 of [WhiteheadRussell] p. 103. Axiom 28 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

Theoremfrege29 40389 Closed form of con3d 155. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))

Theoremfrege30 40390 Commuted, closed form of con3d 155. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))

Theoremaxfrege31 40391 Identical to notnotr 132. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.)
(¬ ¬ 𝜑𝜑)

Axiomax-frege31 40392 𝜑 cannot be denied and (at the same time ) ¬ ¬ 𝜑 affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnotr 132. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)

Theoremfrege32 40393 Deduce con1 148 from con3 156. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))

Theoremfrege33 40394 If 𝜑 or 𝜓 takes place, then 𝜓 or 𝜑 takes place. Identical to con1 148. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → (¬ 𝜓𝜑))

Theoremfrege34 40395 If as a conseqence of the occurrence of the circumstance 𝜑, when the obstacle 𝜓 is removed, 𝜒 takes place, then from the circumstance that 𝜒 does not take place while 𝜑 occurs the occurrence of the obstacle 𝜓 can be inferred. Closed form of con1d 147. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓)))

Theoremfrege35 40396 Commuted, closed form of con1d 147. Proposition 35 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → (¬ 𝜒 → (𝜑𝜓)))

Theoremfrege36 40397 The case in which 𝜓 is denied, ¬ 𝜑 is affirmed, and 𝜑 is affirmed does not occur. If 𝜑 occurs, then (at least) one of the two, 𝜑 or 𝜓, takes place (no matter what 𝜓 might be). Identical to pm2.24 124. Proposition 36 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (¬ 𝜑𝜓))

Theoremfrege37 40398 If 𝜒 is a necessary consequence of the occurrence of 𝜓 or 𝜑, then 𝜒 is a necessary consequence of 𝜑 alone. Similar to a closed form of orcs 872. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → 𝜒) → (𝜑𝜒))

Theoremfrege38 40399 Identical to pm2.21 123. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝜑 → (𝜑𝜓))

Theoremfrege39 40400 Syllogism between pm2.18 128 and pm2.24 124. Proposition 39 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜑) → (¬ 𝜑𝜓))

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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 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 45101-45187
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