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Theorem List for Metamath Proof Explorer - 35001-35100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnndvlem2 35001* Lemma for cnndv 35002. (Contributed by Asger C. Ipsen, 26-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))       𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)
 
Theoremcnndv 35002 There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 34967 and knoppndv 34997. (Contributed by Asger C. Ipsen, 26-Aug-2021.)
𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)
 
21.15  Mathbox for BJ

In this mathbox, we try to respect the ordering of the sections of the main part. There are strengthenings of theorems of the main part, as well as work on reducing axiom dependencies.

 
21.15.1  Propositional calculus

Miscellaneous utility theorems of propositional calculus.

 
21.15.1.1  Derived rules of inference

In this section, we prove a few rules of inference derived from modus ponens ax-mp 5, and which do not depend on any other axioms.

 
Theorembj-mp2c 35003 A double modus ponens inference. Inference associated with mpd 15. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       𝜒
 
Theorembj-mp2d 35004 A double modus ponens inference. Inference associated with mpcom 38. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜓 → (𝜑𝜒))       𝜒
 
21.15.1.2  A syntactic theorem

In this section, we prove a syntactic theorem (bj-0 35005) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 35006) and explain in the comment of that theorem why this phenomenon is unusual.

 
Theorembj-0 35005 A syntactic theorem. See the section comment and the comment of bj-1 35006. The full proof (that is, with the syntactic, non-essential steps) does not appear on this webpage. It has five steps and reads $= wph wps wi wch wi $. The only other syntactic theorems in the main part of set.mm are wel 2107 and weq 1966. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
wff ((𝜑𝜓) → 𝜒)
 
Theorembj-1 35006 In this proof, the use of the syntactic theorem bj-0 35005 allows to reduce the total length by one (non-essential) step. See also the section comment and the comment of bj-0 35005. Since bj-0 35005 is used in a non-essential step, this use does not appear on this webpage (but the present theorem appears on the webpage for bj-0 35005 as a theorem referencing it). The full proof reads $= wph wps wch bj-0 id $. (while, without using bj-0 35005, it would read $= wph wps wi wch wi id $.).

Now we explain why syntactic theorems are not useful in set.mm. Suppose that the syntactic theorem thm-0 proves that PHI is a well-formed formula, and that thm-0 is used to shorten the proof of thm-1. Assume that PHI does have proper non-atomic subformulas (which is not the case of the formula proved by weq 1966 or wel 2107). Then, the proof of thm-1 does not construct all the proper non-atomic subformulas of PHI (if it did, then using thm-0 would not shorten it). Therefore, thm-1 is a special instance of a more general theorem with essentially the same proof. In the present case, bj-1 35006 is a special instance of id 22. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

(((𝜑𝜓) → 𝜒) → ((𝜑𝜓) → 𝜒))
 
21.15.1.3  Minimal implicational calculus
 
Theorembj-a1k 35007 Weakening of ax-1 6. As a consequence, its associated inference is an instance (where we allow extra hypotheses) of ax-1 6. Its commuted form is 2a1 28 (but bj-a1k 35007 does not require ax-2 7). This shortens the proofs of dfwe2 7708 (937>925), ordunisuc2 7780 (789>777), r111 9711 (558>545), smo11 8310 (1176>1164). (Contributed by BJ, 11-Aug-2020.) (Proof modification is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜓)))
 
Theorembj-poni 35008 Inference associated with "pon", pm2.27 42. Its associated inference is ax-mp 5. (Contributed by BJ, 30-Jul-2024.)
𝜑       ((𝜑𝜓) → 𝜓)
 
Theorembj-nnclav 35009 When is substituted for 𝜓, this formula is the Clavius law with a doubly negated consequent, which is therefore a minimalistic tautology. Notice the non-intuitionistic proof from peirce 201 and pm2.27 42 chained using syl 17. (Contributed by BJ, 4-Dec-2023.)
(((𝜑𝜓) → 𝜑) → ((𝜑𝜓) → 𝜓))
 
Theorembj-nnclavi 35010 Inference associated with bj-nnclav 35009. Its associated inference is an instance of syl 17. Notice the non-intuitionistic proof from bj-peircei 35029 and bj-poni 35008. (Contributed by BJ, 30-Jul-2024.)
((𝜑𝜓) → 𝜑)       ((𝜑𝜓) → 𝜓)
 
Theorembj-nnclavc 35011 Commuted form of bj-nnclav 35009. Notice the non-intuitionistic proof from bj-peircei 35029 and imim1i 63. (Contributed by BJ, 30-Jul-2024.) A proof which is shorter when compressed uses embantd 59. (Proof modification is discouraged.)
((𝜑𝜓) → (((𝜑𝜓) → 𝜑) → 𝜓))
 
Theorembj-nnclavci 35012 Inference associated with bj-nnclavc 35011. Its associated inference is an instance of syl 17. Notice the non-intuitionistic proof from peirce 201 and syl 17. (Contributed by BJ, 30-Jul-2024.)
(𝜑𝜓)       (((𝜑𝜓) → 𝜑) → 𝜓)
 
Theorembj-jarrii 35013 Inference associated with jarri 107. Contrary to it, it does not require ax-2 7, but only ax-mp 5 and ax-1 6. (Contributed by BJ, 29-Mar-2020.) (Proof modification is discouraged.)
((𝜑𝜓) → 𝜒)    &   𝜓       𝜒
 
Theorembj-imim21 35014 The propositional function (𝜒 → (. → 𝜃)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
((𝜑𝜓) → ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃))))
 
Theorembj-imim21i 35015 Inference associated with bj-imim21 35014. Its associated inference is syl5 34. (Contributed by BJ, 19-Jul-2019.)
(𝜑𝜓)       ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃)))
 
Theorembj-peircestab 35016 Over minimal implicational calculus, Peirce's law implies the double negation of the stability of any formula (that is the interpretation when is substituted for 𝜓 and for 𝜒). Therefore, the double negation of the stability of any formula is provable in classical refutability calculus. It is also provable in intuitionistic calculus (see iset.mm/bj-nnst) but it is not provable in minimal calculus (see bj-stabpeirce 35017). (Contributed by BJ, 30-Nov-2023.) Axiom ax-3 8 is only used through Peirce's law peirce 201. (Proof modification is discouraged.)
(((((𝜑𝜓) → 𝜒) → 𝜑) → 𝜒) → 𝜒)
 
Theorembj-stabpeirce 35017 This minimal implicational calculus tautology is used in the following argument: When 𝜑, 𝜓, 𝜒, 𝜃, 𝜏 are replaced respectively by (𝜑 → ⊥), ⊥, 𝜑, ⊥, ⊥, the antecedent becomes ¬ ¬ (¬ ¬ 𝜑𝜑), that is, the double negation of the stability of 𝜑. If that statement were provable in minimal calculus, then, since plays no particular role in minimal calculus, also the statement with 𝜓 in place of would be provable. The corresponding consequent is (((𝜓𝜑) → 𝜓) → 𝜓), that is, the non-intuitionistic Peirce law. Therefore, the double negation of the stability of any formula is not provable in minimal calculus. However, it is provable both in intuitionistic calculus (see iset.mm/bj-nnst) and in classical refutability calculus (see bj-peircestab 35016). (Contributed by BJ, 30-Nov-2023.) (Revised by BJ, 30-Jul-2024.) (Proof modification is discouraged.)
(((((𝜑𝜓) → 𝜒) → 𝜃) → 𝜏) → (((𝜓𝜒) → 𝜃) → 𝜏))
 
21.15.1.4  Positive calculus

Positive calculus is understood to be intuitionistic.

 
Theorembj-syl66ib 35018 A mixed syllogism inference derived from syl6ib 250. In addition to bj-dvelimdv1 35318, it can also shorten alexsubALTlem4 23401 (4821>4812), supsrlem 11047 (2868>2863). (Contributed by BJ, 20-Oct-2021.)
(𝜑 → (𝜓𝜃))    &   (𝜃𝜏)    &   (𝜏𝜒)       (𝜑 → (𝜓𝜒))
 
Theorembj-orim2 35019 Proof of orim2 966 from the axiomatic definition of disjunction (olc 866, orc 865, jao 959) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))
 
Theorembj-currypeirce 35020 Curry's axiom curryax 892 (a non-intuitionistic positive statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 201 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the elimination axiom jao 959 via its inference form jaoi 855; the introduction axioms olc 866 and orc 865 are not needed). Note that this theorem shows that actually, the standard instance of curryax 892 implies the standard instance of peirce 201, which is not the case for the converse bj-peircecurry 35021. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∨ (𝜑𝜓)) → (((𝜑𝜓) → 𝜑) → 𝜑))
 
Theorembj-peircecurry 35021 Peirce's axiom peirce 201 implies Curry's axiom curryax 892 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the introduction axioms olc 866 and orc 865; the elimination axiom jao 959 is not needed). See bj-currypeirce 35020 for the converse. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ∨ (𝜑𝜓))
 
Theorembj-animbi 35022 Conjunction in terms of implication and biconditional. Note that the proof is intuitionistic (use of ax-3 8 comes from the unusual definition of the biconditional in set.mm). (Contributed by BJ, 23-Sep-2023.)
((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
 
Theorembj-currypara 35023 Curry's paradox. Note that the proof is intuitionistic (use ax-3 8 comes from the unusual definition of the biconditional in set.mm). The paradox comes from the case where 𝜑 is the self-referential sentence "If this sentence is true, then 𝜓", so that one can prove everything. Therefore, a consistent system cannot allow the formation of such self-referential sentences. This has lead to the study of logics rejecting contraction pm2.43 56, such as affine logic and linear logic. (Contributed by BJ, 23-Sep-2023.)
((𝜑 ↔ (𝜑𝜓)) → 𝜓)
 
21.15.1.5  Implication and negation
 
Theorembj-con2com 35024 A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
(𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))
 
Theorembj-con2comi 35025 Inference associated with bj-con2com 35024. Its associated inference is mt2 199. TODO: when in the main part, add to mt2 199 that it is the inference associated with bj-con2comi 35025. (Contributed by BJ, 19-Mar-2020.)
𝜑       ((𝜓 → ¬ 𝜑) → ¬ 𝜓)
 
Theorembj-pm2.01i 35026 Inference associated with the weak Clavius law pm2.01 188. (Contributed by BJ, 30-Mar-2020.)
(𝜑 → ¬ 𝜑)        ¬ 𝜑
 
Theorembj-nimn 35027 If a formula is true, then it does not imply its negation. (Contributed by BJ, 19-Mar-2020.) A shorter proof is possible using id 22 and jc 161, however, the present proof uses theorems that are more basic than jc 161. (Proof modification is discouraged.)
(𝜑 → ¬ (𝜑 → ¬ 𝜑))
 
Theorembj-nimni 35028 Inference associated with bj-nimn 35027. (Contributed by BJ, 19-Mar-2020.)
𝜑        ¬ (𝜑 → ¬ 𝜑)
 
Theorembj-peircei 35029 Inference associated with peirce 201. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜑)       𝜑
 
Theorembj-looinvi 35030 Inference associated with looinv 202. Its associated inference is bj-looinvii 35031. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜓)       ((𝜓𝜑) → 𝜑)
 
Theorembj-looinvii 35031 Inference associated with bj-looinvi 35030. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜓)    &   (𝜓𝜑)       𝜑
 
Theorembj-mt2bi 35032 Version of mt2 199 where the major premise is a biconditional. Another proof is also possible via con2bii 357 and mpbi 229. The current mt2bi 363 should be relabeled, maybe to imfal. (Contributed by BJ, 5-Oct-2024.)
𝜑    &   (𝜓 ↔ ¬ 𝜑)        ¬ 𝜓
 
Theorembj-ntrufal 35033 The negation of a theorem is equivalent to false. This can shorten dfnul2 4285. (Contributed by BJ, 5-Oct-2024.)
𝜑       𝜑 ↔ ⊥)
 
Theorembj-fal 35034 Shortening of fal 1555 using bj-mt2bi 35032. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) (Proof modification is discouraged.)
¬ ⊥
 
21.15.1.6  Disjunction

A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 558 and pm4.72 948. See also biort 934 and biorf 935.

 
Theorembj-jaoi1 35035 Shortens orfa2 36545 (58>53), pm1.2 902 (20>18), pm1.2 902 (20>18), pm2.4 905 (31>25), pm2.41 906 (31>25), pm2.42 941 (38>32), pm3.2ni 879 (43>39), pm4.44 995 (55>51). (Contributed by BJ, 30-Sep-2019.)
(𝜑𝜓)       ((𝜑𝜓) → 𝜓)
 
Theorembj-jaoi2 35036 Shortens consensus 1051 (110>106), elnn0z 12512 (336>329), pm1.2 902 (20>19), pm3.2ni 879 (43>39), pm4.44 995 (55>51). (Contributed by BJ, 30-Sep-2019.)
(𝜑𝜓)       ((𝜓𝜑) → 𝜓)
 
21.15.1.7  Logical equivalence

A few other characterizations of the bicondional. The inter-definability of logical connectives offers many ways to express a given statement. Some useful theorems in this regard are df-or 846, df-an 397, pm4.64 847, imor 851, pm4.62 854 through pm4.67 399, and, for the De Morgan laws, ianor 980 through pm4.57 989.

 
Theorembj-dfbi4 35037 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))
 
Theorembj-dfbi5 35038 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) → (𝜑𝜓)))
 
Theorembj-dfbi6 35039 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))
 
Theorembj-bijust0ALT 35040 Alternate proof of bijust0 203; shorter but using additional intermediate results. (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Revised by BJ, 19-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ((𝜑𝜑) → ¬ (𝜑𝜑))
 
Theorembj-bijust00 35041 A self-implication does not imply the negation of a self-implication. Most general theorem of which bijust 204 is an instance (bijust0 203 and bj-bijust0ALT 35040 are therefore also instances of it). (Contributed by BJ, 7-Sep-2022.)
¬ ((𝜑𝜑) → ¬ (𝜓𝜓))
 
21.15.1.8  The conditional operator for propositions
 
Theorembj-consensus 35042 Version of consensus 1051 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1051, using only binary connectives, and hinting at the fact that it is a Boolean algebra identity, like the absorption identities.) (Contributed by BJ, 30-Sep-2019.)
((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
 
Theorembj-consensusALT 35043 Alternate proof of bj-consensus 35042. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))
 
Theorembj-df-ifc 35044* Candidate definition for the conditional operator for classes. This is in line with the definition of a class as the extension of a predicate in df-clab 2714. We reprove the current df-if 4487 from it in bj-dfif 35045. (Contributed by BJ, 20-Sep-2019.) (Proof modification is discouraged.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}
 
Theorembj-dfif 35045* Alternate definition of the conditional operator for classes, which used to be the main definition. (Contributed by BJ, 26-Dec-2023.) (Proof modification is discouraged.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}
 
Theorembj-ififc 35046 A biconditional connecting the conditional operator for propositions and the conditional operator for classes. Note that there is no sethood hypothesis on 𝑋: it is implied by either side. (Contributed by BJ, 24-Sep-2019.) Generalize statement from setvar 𝑥 to class 𝑋. (Revised by BJ, 26-Dec-2023.)
(𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋𝐴, 𝑋𝐵))
 
21.15.1.9  Propositional calculus: miscellaneous

Miscellaneous theorems of propositional calculus.

 
Theorembj-imbi12 35047 Uncurried (imported) form of imbi12 346. (Contributed by BJ, 6-May-2019.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))
 
Theorembj-biorfi 35048 This should be labeled "biorfi" while the current biorfi 937 should be labeled "biorfri". The dual of biorf 935 is not biantr 804 but iba 528 (and ibar 529). So there should also be a "biorfr". (Note that these four statements can actually be strengthened to biconditionals.) (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
¬ 𝜑       (𝜓 ↔ (𝜑𝜓))
 
Theorembj-falor 35049 Dual of truan 1552 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (⊥ ∨ 𝜑))
 
Theorembj-falor2 35050 Dual of truan 1552. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
((⊥ ∨ 𝜑) ↔ 𝜑)
 
Theorembj-bibibi 35051 A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (𝜓 ↔ (𝜑𝜓)))
 
Theorembj-imn3ani 35052 Duplication of bnj1224 33413. Three-fold version of imnani 401. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 22-Oct-2019.) (Proof modification is discouraged.)
¬ (𝜑𝜓𝜒)       ((𝜑𝜓) → ¬ 𝜒)
 
Theorembj-andnotim 35053 Two ways of expressing a certain ternary connective. Note the respective positions of the three formulas on each side of the biconditional. (Contributed by BJ, 6-Oct-2018.)
(((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
 
Theorembj-bi3ant 35054 This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
(𝜑 → (𝜓𝜒))       (((𝜃𝜏) → 𝜑) → (((𝜏𝜃) → 𝜓) → ((𝜃𝜏) → 𝜒)))
 
Theorembj-bisym 35055 This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.)
(((𝜑𝜓) → (𝜒𝜃)) → (((𝜓𝜑) → (𝜃𝜒)) → ((𝜑𝜓) → (𝜒𝜃))))
 
Theorembj-bixor 35056 Equivalence of two ternary operations. Note the identical order and parenthesizing of the three arguments in both expressions. (Contributed by BJ, 31-Dec-2023.)
((𝜑 ↔ (𝜓𝜒)) ↔ (𝜑 ⊻ (𝜓𝜒)))
 
21.15.2  Modal logic

In this section, we prove some theorems related to modal logic. For modal logic, we refer to https://en.wikipedia.org/wiki/Kripke_semantics, https://en.wikipedia.org/wiki/Modal_logic and https://plato.stanford.edu/entries/logic-modal/.

Monadic first-order logic (i.e., with quantification over only one variable) is bi-interpretable with modal logic, by mapping 𝑥 to "necessity" (generally denoted by a box) and 𝑥 to "possibility" (generally denoted by a diamond). Therefore, we use these quantifiers so as not to introduce new symbols. (To be strictly within modal logic, we should add disjoint variable conditions between 𝑥 and any other metavariables appearing in the statements.)

For instance, ax-gen 1797 corresponds to the necessitation rule of modal logic, and ax-4 1811 corresponds to the distributivity axiom (K) of modal logic, also called the Kripke scheme. Modal logics satisfying these rule and axiom are called "normal modal logics", of which the most important modal logics are.

The minimal normal modal logic is also denoted by (K). Here are a few normal modal logics with their axiomatizations (on top of (K)): (K) axiomatized by no supplementary axioms; (T) axiomatized by the axiom T; (K4) axiomatized by the axiom 4; (S4) axiomatized by the axioms T,4; (S5) axiomatized by the axioms T,5 or D,B,4; (GL) axiomatized by the axiom GL.

The last one, called Gödel–Löb logic or provability logic, is important because it describes exactly the properties of provability in Peano arithmetic, as proved by Robert Solovay. See for instance https://plato.stanford.edu/entries/logic-provability/ 1811. A basic result in this logic is bj-gl4 35060.

 
Theorembj-axdd2 35057 This implication, proved using only ax-gen 1797 and ax-4 1811 on top of propositional calculus (hence holding, up to the standard interpretation, in any normal modal logic), shows that the axiom scheme 𝑥 implies the axiom scheme (∀𝑥𝜑 → ∃𝑥𝜑). These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axd2d 35058. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))
 
Theorembj-axd2d 35058 This implication, proved using only ax-gen 1797 on top of propositional calculus (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme (∀𝑥𝜑 → ∃𝑥𝜑) implies the axiom scheme 𝑥. These correspond to the modal axiom (D), and in predicate calculus, they assert that the universe of discourse is nonempty. For the converse, see bj-axdd2 35057. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)
 
Theorembj-axtd 35059 This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme (∀𝑥𝜑𝜑) (modal T) implies the axiom scheme (∀𝑥𝜑 → ∃𝑥𝜑) (modal D). See also bj-axdd2 35057 and bj-axd2d 35058. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))
 
Theorembj-gl4 35060 In a normal modal logic, the modal axiom GL implies the modal axiom (4). Translated to first-order logic, Axiom GL reads (∀𝑥(∀𝑥𝜑𝜑) → ∀𝑥𝜑). Note that the antecedent of bj-gl4 35060 is an instance of the axiom GL, with 𝜑 replaced by (∀𝑥𝜑𝜑), which is a modality sometimes called the "strong necessity" of 𝜑. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)) → ∀𝑥(∀𝑥𝜑𝜑)) → (∀𝑥𝜑 → ∀𝑥𝑥𝜑))
 
Theorembj-axc4 35061 Over minimal calculus, the modal axiom (4) (hba1 2289) and the modal axiom (K) (ax-4 1811) together imply axc4 2314. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝜑 → ∀𝑥𝑥𝜑) → ((∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓)) → (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))))
 
21.15.3  Provability logic

In this section, we assume that, on top of propositional calculus, there is given a provability predicate Prv satisfying the three axioms ax-prv1 35063 and ax-prv2 35064 and ax-prv3 35065. Note the similarity with ax-gen 1797, ax-4 1811 and hba1 2289 respectively. These three properties of Prv are often called the Hilbert–Bernays–Löb derivability conditions, or the Hilbert–Bernays provability conditions.

This corresponds to the modal logic (K4) (see previous section for modal logic). The interpretation of provability logic is the following: we are given a background first-order theory T, the wff Prv 𝜑 means "𝜑 is provable in T", and the turnstile indicates provability in T.

Beware that "provability logic" often means (K) augmented with the Gödel–Löb axiom GL, which we do not assume here (at least for the moment). See for instance https://plato.stanford.edu/entries/logic-provability/ 2289.

Provability logic is worth studying because whenever T is a first-order theory containing Robinson arithmetic (a fragment of Peano arithmetic), one can prove (using Gödel numbering, and in the much weaker primitive recursive arithmetic) that there exists in T a provability predicate Prv satisfying the above three axioms. (We do not construct this predicate in this section; this is still a project.)

The main theorems of this section are the "easy parts" of the proofs of Gödel's second incompleteness theorem (bj-babygodel 35068) and Löb's theorem (bj-babylob 35069). See the comments of these theorems for details.

 
Syntaxcprvb 35062 Syntax for the provability predicate.
wff Prv 𝜑
 
Axiomax-prv1 35063 First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
𝜑       Prv 𝜑
 
Axiomax-prv2 35064 Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv (𝜑𝜓) → (Prv 𝜑 → Prv 𝜓))
 
Axiomax-prv3 35065 Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv 𝜑 → Prv Prv 𝜑)
 
Theoremprvlem1 35066 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑𝜓)       (Prv 𝜑 → Prv 𝜓)
 
Theoremprvlem2 35067 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑 → (𝜓𝜒))       (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))
 
Theorembj-babygodel 35068 See the section header comments for the context.

The first hypothesis reads "𝜑 is true if and only if it is not provable in T" (and having this first hypothesis means that we can prove this fact in T). The wff 𝜑 is a formal version of the sentence "This sentence is not provable". The hard part of the proof of Gödel's theorem is to construct such a 𝜑, called a "Gödel–Rosser sentence", for a first-order theory T which is effectively axiomatizable and contains Robinson arithmetic, through Gödel diagonalization (this can be done in primitive recursive arithmetic). The second hypothesis means that is not provable in T, that is, that the theory T is consistent (and having this second hypothesis means that we can prove in T that the theory T is consistent). The conclusion is the falsity, so having the conclusion means that T can prove the falsity, that is, T is inconsistent.

Therefore, taking the contrapositive, this theorem expresses that if a first-order theory is consistent (and one can prove in it that some formula is true if and only if it is not provable in it), then this theory does not prove its own consistency.

This proof is due to George Boolos, Gödel's Second Incompleteness Theorem Explained in Words of One Syllable, Mind, New Series, Vol. 103, No. 409 (January 1994), pp. 1--3.

(Contributed by BJ, 3-Apr-2019.)

(𝜑 ↔ ¬ Prv 𝜑)    &    ¬ Prv ⊥       
 
Theorembj-babylob 35069 See the section header comments for the context, as well as the comments for bj-babygodel 35068.

Löb's theorem when the Löb sentence is given as a hypothesis (the hard part of the proof of Löb's theorem is to construct this Löb sentence; this can be done, using Gödel diagonalization, for any first-order effectively axiomatizable theory containing Robinson arithmetic). More precisely, the present theorem states that if a first-order theory proves that the provability of a given sentence entails its truth (and if one can construct in this theory a provability predicate and a Löb sentence, given here as the first hypothesis), then the theory actually proves that sentence.

See for instance, Eliezer Yudkowsky, The Cartoon Guide to Löb's Theorem (available at http://yudkowsky.net/rational/lobs-theorem/ 35068).

(Contributed by BJ, 20-Apr-2019.)

(𝜓 ↔ (Prv 𝜓𝜑))    &   (Prv 𝜑𝜑)       𝜑
 
Theorembj-godellob 35070 Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 35068 and bj-babylob 35069 for details). (Contributed by BJ, 20-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ ¬ Prv 𝜑)    &    ¬ Prv ⊥       
 
21.15.4  First-order logic

Utility lemmas or strengthenings of theorems in the main part (biconditional or closed forms, or fewer disjoint variable conditions, or disjoint variable conditions replaced with nonfreeness hypotheses...). Sorted in the same order as in the main part.

 
21.15.4.1  Adding ax-gen
 
Theorembj-genr 35071 Generalization rule on the right conjunct. See 19.28 2221. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (𝜑 ∧ ∀𝑥𝜓)
 
Theorembj-genl 35072 Generalization rule on the left conjunct. See 19.27 2220. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)
 
Theorembj-genan 35073 Generalization rule on a conjunction. Forward inference associated with 19.26 1873. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑 ∧ ∀𝑥𝜓)
 
Theorembj-mpgs 35074 From a closed form theorem (the major premise) with an antecedent in the "strong necessity" modality (in the language of modal logic), deduce the inference 𝜑𝜓. Strong necessity is stronger than necessity, and equivalent to it when sp 2176 (modal T) is available. Therefore, this theorem is stronger than mpg 1799 when sp 2176 is not available. (Contributed by BJ, 1-Nov-2023.)
𝜑    &   ((𝜑 ∧ ∀𝑥𝜑) → 𝜓)       𝜓
 
21.15.4.2  Adding ax-4
 
Theorembj-2alim 35075 Closed form of 2alimi 1814. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓))
 
Theorembj-2exim 35076 Closed form of 2eximi 1838. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓))
 
Theorembj-alanim 35077 Closed form of alanimi 1818. (Contributed by BJ, 6-May-2019.)
(∀𝑥((𝜑𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))
 
Theorembj-2albi 35078 Closed form of 2albii 1822. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓))
 
Theorembj-notalbii 35079 Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4334 (103>94), ballotlem2 33088 (2655>2648), bnj1143 33402 (522>519), hausdiag 22996 (2119>2104). (Contributed by BJ, 17-Jul-2021.)
(𝜑𝜓)       (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)
 
Theorembj-2exbi 35080 Closed form of 2exbii 1851. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓))
 
Theorembj-3exbi 35081 Closed form of 3exbii 1852. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦𝑧(𝜑𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))
 
Theorembj-sylgt2 35082 Uncurried (imported) form of sylgt 1824. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜓𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))
 
Theorembj-alrimg 35083 The general form of the *alrim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in an antecedent, one can deduce from the universally quantified implication an implication where the consequent is universally quantified. Dual of bj-exlimg 35087. (Contributed by BJ, 9-Dec-2023.)
((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓𝜒) → (𝜑 → ∀𝑥𝜒)))
 
Theorembj-alrimd 35084 A slightly more general alrimd 2208. A common usage will have 𝜑 substituted for 𝜓 and 𝜒 substituted for 𝜃, giving a form closer to alrimd 2208. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑥𝜓)    &   (𝜑 → (𝜒 → ∀𝑥𝜃))    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜒 → ∀𝑥𝜏))
 
Theorembj-sylget 35085 Dual statement of sylgt 1824. Closed form of bj-sylge 35088. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜒𝜑) → ((∃𝑥𝜑𝜓) → (∃𝑥𝜒𝜓)))
 
Theorembj-sylget2 35086 Uncurried (imported) form of bj-sylget 35085. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜑𝜓) ∧ (∃𝑥𝜓𝜒)) → (∃𝑥𝜑𝜒))
 
Theorembj-exlimg 35087 The general form of the *exlim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in a consequent, one can deduce from the universally quantified implication an implication where the antecedent is existentially quantified. Dual of bj-alrimg 35083. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑𝜓) → (∀𝑥(𝜒𝜑) → (∃𝑥𝜒𝜓)))
 
Theorembj-sylge 35088 Dual statement of sylg 1825 (the final "e" in the label stands for "existential (version of sylg 1825)". Variant of exlimih 2285. (Contributed by BJ, 25-Dec-2023.)
(∃𝑥𝜑𝜓)    &   (𝜒𝜑)       (∃𝑥𝜒𝜓)
 
Theorembj-exlimd 35089 A slightly more general exlimd 2211. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2211. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑥𝜓)    &   (𝜑 → (∃𝑥𝜃𝜏))    &   (𝜓 → (𝜒𝜃))       (𝜑 → (∃𝑥𝜒𝜏))
 
Theorembj-nfimexal 35090 A weak from of nonfreeness in either an antecedent or a consequent implies that a universally quantified implication is equivalent to the associated implication where the antecedent is existentially quantified and the consequent is universally quantified. The forward implication always holds (this is 19.38 1841) and the converse implication is the join of instances of bj-alrimg 35083 and bj-exlimg 35087 (see 19.38a 1842 and 19.38b 1843). TODO: prove a version where the antecedents use the nonfreeness quantifier. (Contributed by BJ, 9-Dec-2023.)
(((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
 
Theorembj-alexim 35091 Closed form of aleximi 1834. Note: this proof is shorter, so aleximi 1834 could be deduced from it (exim 1836 would have to be proved first, see bj-eximALT 35105 but its proof is shorter (currently almost a subproof of aleximi 1834)). (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))
 
Theorembj-nexdh 35092 Closed form of nexdh 1868 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))
 
Theorembj-nexdh2 35093 Uncurried (imported) form of bj-nexdh 35092. (Contributed by BJ, 6-May-2019.)
((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))
 
Theorembj-hbxfrbi 35094 Closed form of hbxfrbi 1827. Note: it is less important than nfbiit 1853. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht 35206) in order not to require sp 2176 (modal T). See bj-hbyfrbi 35095 for its version with existential quantifiers. (Contributed by BJ, 6-May-2019.)
(((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))
 
Theorembj-hbyfrbi 35095 Version of bj-hbxfrbi 35094 with existential quantifiers. (Contributed by BJ, 23-Aug-2023.)
(((𝜑𝜓) ∧ ∀𝑥(𝜑𝜓)) → ((∃𝑥𝜑𝜑) ↔ (∃𝑥𝜓𝜓)))
 
Theorembj-exalim 35096 Distribute quantifiers over a nested implication.

This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1913. I propose to move to the main part: bj-exalim 35096, bj-exalimi 35097, bj-exalims 35098, bj-exalimsi 35099, bj-ax12i 35101, bj-ax12wlem 35108, bj-ax12w 35141. A new label is needed for bj-ax12i 35101 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to 𝑥 in speimfw 1967 and spimfw 1969 (other spim* theorems use 𝑥 and very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 8-Nov-2021.)

(∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))
 
Theorembj-exalimi 35097 An inference for distributing quantifiers over a nested implication. The canonical derivation from its closed form bj-exalim 35096 (using mpg 1799) has fewer essential steps, but more steps in total (yielding a longer compressed proof). (Almost) the general statement that speimfw 1967 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))
 
Theorembj-exalims 35098 Distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1969 proves. (Contributed by BJ, 29-Sep-2019.)
(∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓𝜒)))
 
Theorembj-exalimsi 35099 An inference for distributing quantifiers over a nested implication. (Almost) the general statement that spimfw 1969 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓𝜒))
 
Theorembj-ax12ig 35100 A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 35101. (Contributed by BJ, 19-Dec-2020.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))
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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 47201-47242
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