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Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremknoppndvlem22 32801* Lemma for knoppndv 32802. (Contributed by Asger C. Ipsen, 19-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐻 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ ((𝑎𝐻𝐻𝑏) ∧ ((𝑏𝑎) < 𝐷𝑎𝑏) ∧ 𝐸 ≤ ((abs‘((𝑊𝑏) − (𝑊𝑎))) / (𝑏𝑎))))

Theoremknoppndv 32802* The continuous nowhere differentiable function 𝑊 ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, nowhere differentiable. (Contributed by Asger C. Ipsen, 19-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 1 < (𝑁 · (abs‘𝐶)))       (𝜑 → dom (ℝ D 𝑊) = ∅)

Theoremknoppf 32803* Knopp's function is a function. (Contributed by Asger C. Ipsen, 25-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝐶 ∈ (-1(,)1))    &   (𝜑𝑁 ∈ ℕ)       (𝜑𝑊:ℝ⟶ℝ)

Theoremknoppcn2 32804* Variant of knoppcn 32771 with different codomain. (Contributed by Asger C. Ipsen, 25-Aug-2021.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹𝑤)‘𝑖))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐶 ∈ (-1(,)1))       (𝜑𝑊 ∈ (ℝ–cn→ℝ))

Theoremcnndvlem1 32805* Lemma for cnndv 32807. (Contributed by Asger C. Ipsen, 25-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))       (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)

Theoremcnndvlem2 32806* Lemma for cnndv 32807. (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 32807 There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 32771 and knoppndv 32802. (Contributed by Asger C. Ipsen, 26-Aug-2021.)
𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)

20.14  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.

20.14.1  Propositional calculus

Miscellaneous utility theorems of propositional calculus.

20.14.1.1  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 32808 A double modus ponens inference. Inference associated with mpd 15. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       𝜒

Theorembj-mp2d 32809 A double modus ponens inference. Inference associated with mpcom 38. (Contributed by BJ, 24-Sep-2019.)
𝜑    &   (𝜑𝜓)    &   (𝜓 → (𝜑𝜒))       𝜒

20.14.1.2  A syntactic theorem

In this section, we prove a syntactic theorem (bj-0 32810) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 32811) and explain in the comment of that theorem why this phenomenon is unusual.

Theorembj-0 32810 A syntactic theorem. See the section comment and the comment of bj-1 32811. 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 2128 and weq 2028. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
wff ((𝜑𝜓) → 𝜒)

Theorembj-1 32811 In this proof, the use of the syntactic theorem bj-0 32810 allows to reduce the total length by one (non-essential) step. See also the section comment and the comment of bj-0 32810. Since bj-0 32810 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 32810 as a theorem referencing it). The full proof reads \$= wph wps wch bj-0 id \$. (while, without using bj-0 32810, 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 2028 or wel 2128). 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 32811 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))

20.14.1.3  Minimal implicational calculus

Theorembj-a1k 32812 Weakening of ax-1 6. This shortens the proofs of dfwe2 7134 (937>925), ordunisuc2 7197 (789>777), r111 8799 (558>545), smo11 7618 (1176>1164). (Contributed by BJ, 11-Aug-2020.)
(𝜑 → (𝜓 → (𝜒𝜓)))

Theorembj-jarri 32813 Inference associated with jarr 106. Its associated inference is bj-jarrii 32814. (Contributed by BJ, 29-Mar-2020.)
((𝜑𝜓) → 𝜒)       (𝜓𝜒)

Theorembj-jarrii 32814 Inference associated with bj-jarri 32813. (Contributed by BJ, 29-Mar-2020.)
((𝜑𝜓) → 𝜒)    &   𝜓       𝜒

Theorembj-imim2ALT 32815 More direct proof of imim2 58. Note that imim2i 16 and imim2d 57 can be proved as usual from this closed form (i.e., using ax-mp 5 and syl 17 respectively). (Contributed by BJ, 19-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theorembj-imim21 32816 The propositional function (𝜒 → (. → 𝜃)) is decreasing. (Contributed by BJ, 19-Jul-2019.)
((𝜑𝜓) → ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃))))

Theorembj-imim21i 32817 Inference associated with bj-imim21 32816. Its associated inference is syl5 34. (Contributed by BJ, 19-Jul-2019.)
(𝜑𝜓)       ((𝜒 → (𝜓𝜃)) → (𝜒 → (𝜑𝜃)))

20.14.1.4  Positive calculus

Theorembj-orim2 32818 Proof of orim2 922 from the axiomatic definition of disjunction (olc 398, orc 399, jao 535) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))

Theorembj-curry 32819 A non-intuitionistic positive statement, sometimes called a paradox of material implication. Sometimes called Curry's axiom. (Contributed by BJ, 4-Apr-2021.)
(𝜑 ∨ (𝜑𝜓))

Theorembj-peirce 32820 Proof of peirce 193 from minimal implicational calculus, the axiomatic definition of disjunction (olc 398, orc 399, jao 535), and Curry's axiom bj-curry 32819. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜑) → 𝜑)

Theorembj-currypeirce 32821 Curry's axiom (a non-intuitionistic statement sometimes called a paradox of material implication) implies Peirce's axiom peirce 193 over minimal implicational calculus and the axiomatic definition of disjunction (olc 398, orc 399, jao 535). A shorter proof from bj-orim2 32818, pm1.2 536, syl6com 37 is possible if we accept to use pm1.2 536, itself a direct consequence of jao 535. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∨ (𝜑𝜓)) → (((𝜑𝜓) → 𝜑) → 𝜑))

Theorembj-peircecurry 32822 Peirce's axiom peirce 193 implies Curry's axiom over minimal implicational calculus and the axiomatic definition of disjunction (olc 398, orc 399, jao 535). See comment of bj-currypeirce 32821. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ∨ (𝜑𝜓))

20.14.1.5  Implication and negation

Theorempm4.81ALT 32823 Alternate proof of pm4.81 380. (Contributed by BJ, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((¬ 𝜑𝜑) ↔ 𝜑)

Theorembj-con4iALT 32824 Alternate proof of con4i 113. Probably the original proof. (Contributed by BJ, 29-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝜑 → ¬ 𝜓)       (𝜓𝜑)

Theorembj-con2com 32825 A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.)
(𝜑 → ((𝜓 → ¬ 𝜑) → ¬ 𝜓))

Theorembj-con2comi 32826 Inference associated with bj-con2com 32825. Its associated inference is mt2 191. TODO: when in the main part, add to mt2 191 that it is the inference associated with bj-con2comi 32826. (Contributed by BJ, 19-Mar-2020.)
𝜑       ((𝜓 → ¬ 𝜑) → ¬ 𝜓)

Theorembj-pm2.01i 32827 Inference associated with pm2.01 180. (Contributed by BJ, 30-Mar-2020.)
(𝜑 → ¬ 𝜑)        ¬ 𝜑

Theorembj-nimn 32828 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 159, however, the present proof uses theorems that are more basic than jc 159. (Proof modification is discouraged.)
(𝜑 → ¬ (𝜑 → ¬ 𝜑))

Theorembj-nimni 32829 Inference associated with bj-nimn 32828. (Contributed by BJ, 19-Mar-2020.)
𝜑        ¬ (𝜑 → ¬ 𝜑)

Theorembj-peircei 32830 Inference associated with peirce 193. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜑)       𝜑

Theorembj-looinvi 32831 Inference associated with looinv 194. Its associated inference is bj-looinvii 32832. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜓)       ((𝜓𝜑) → 𝜑)

Theorembj-looinvii 32832 Inference associated with bj-looinvi 32831. (Contributed by BJ, 30-Mar-2020.)
((𝜑𝜓) → 𝜓)    &   (𝜓𝜑)       𝜑

20.14.1.6  Disjunction

A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 665 and pm4.72 956. See also biort 974 and biorf 419.

Theorembj-jaoi1 32833 Shortens orfa2 34169 (58>53), pm1.2 536 (20>18), pm1.2 536 (20>18), pm2.4 600 (31>25), pm2.41 598 (31>25), pm2.42 599 (38>32), pm3.2ni 935 (43>39), pm4.44 602 (55>51). (Contributed by BJ, 30-Sep-2019.)
(𝜑𝜓)       ((𝜑𝜓) → 𝜓)

Theorembj-jaoi2 32834 Shortens consensus 1041 (110>106), elnn0z 11553 (336>329), pm1.2 536 (20>19), pm3.2ni 935 (43>39), pm4.44 602 (55>51). (Contributed by BJ, 30-Sep-2019.)
(𝜑𝜓)       ((𝜓𝜑) → 𝜓)

20.14.1.7  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 384, df-an 385, pm4.64 386, imor 427, pm4.62 434 through pm4.67 443, and, for the De Morgan laws, ianor 510 through pm4.57 519.

Theorembj-dfbi4 32835 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Theorembj-dfbi5 32836 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) → (𝜑𝜓)))

Theorembj-dfbi6 32837 Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜑𝜓)))

Theorembj-bijust0 32838 The general statement that bijust 195 proves (with a shorter proof). (Contributed by NM, 11-May-1999.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Revised by BJ, 19-Mar-2020.)
¬ ((𝜑𝜑) → ¬ (𝜑𝜑))

20.14.1.8  The conditional operator for propositions

Theorembj-consensus 32839 Version of consensus 1041 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1041, 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 32840 Alternate proof of bj-consensus 32839. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((if-(𝜑, 𝜓, 𝜒) ∨ (𝜓𝜒)) ↔ if-(𝜑, 𝜓, 𝜒))

Theorembj-dfifc2 32841* This should be the alternate definition of "ifc" if "if-" enters the main part. (Contributed by BJ, 20-Sep-2019.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}

Theorembj-df-ifc 32842* The definition of "ifc" if "if-" enters the main part. This is in line with the definition of a class as the extension of a predicate in df-clab 2735. (Contributed by BJ, 20-Sep-2019.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ if-(𝜑, 𝑥𝐴, 𝑥𝐵)}

Theorembj-ififc 32843* A theorem linking if- and if. (Contributed by BJ, 24-Sep-2019.)
(𝑥 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑥𝐴, 𝑥𝐵))

20.14.1.9  Propositional calculus: miscellaneous

Miscellaneous theorems of propositional calculus.

Theorembj-imbi12 32844 Uncurried (imported) form of imbi12 335. (Contributed by BJ, 6-May-2019.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))

Theorembj-biorfi 32845 This should be labeled "biorfi" while the current biorfi 421 should be labeled "biorfri". The dual of biorf 419 is not biantr 1010 but iba 525 (and ibar 526). 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 32846 Dual of truan 1638 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (⊥ ∨ 𝜑))

Theorembj-falor2 32847 Dual of truan 1638. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
((⊥ ∨ 𝜑) ↔ 𝜑)

Theorembj-bibibi 32848 A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.)
(𝜑 ↔ (𝜓 ↔ (𝜑𝜓)))

Theorembj-imn3ani 32849 Duplication of bnj1224 31150. Three-fold version of imnani 438. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 22-Oct-2019.) (Proof modification is discouraged.)
¬ (𝜑𝜓𝜒)       ((𝜑𝜓) → ¬ 𝜒)

Theorembj-andnotim 32850 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 32851 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 32852 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))))

20.14.2  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 dv conditions between 𝑥 and any other metavariables appearing in the statements.)

For instance, ax-gen 1859 corresponds to the necessitation rule of modal logic, and ax-4 1874 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/. A basic result in this logic is bj-gl4 32857.

Theorembj-axdd2 32853 This implication, proved using only ax-gen 1859 and ax-4 1874 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 32854. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))

Theorembj-axd2d 32854 This implication, proved using only ax-gen 1859 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 32853. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥⊤ → ∃𝑥⊤) → ∃𝑥⊤)

Theorembj-axtd 32855 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 32853 and bj-axd2d 32854. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥 ¬ 𝜑 → ¬ 𝜑) → ((∀𝑥𝜑𝜑) → (∀𝑥𝜑 → ∃𝑥𝜑)))

Theorembj-gl4lem 32856 Lemma for bj-gl4 32857. Note that this proof holds in the modal logic (K). (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝜑 → ∀𝑥(∀𝑥(∀𝑥𝜑𝜑) → (∀𝑥𝜑𝜑)))

Theorembj-gl4 32857 In a normal modal logic, the modal axiom GL implies the modal axiom (4). Note that the antecedent of bj-gl4 32857 is an instance of the axiom GL, with 𝜑 replaced by (∀𝑥𝜑𝜑), 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 32858 Over minimal calculus, the modal axiom (4) (hba1 2286) and the modal axiom (K) (ax-4 1874) together imply axc4 2265. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝜑 → ∀𝑥𝑥𝜑) → ((∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓)) → (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))))

20.14.3  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 32860 and ax-prv2 32861 and ax-prv3 32862. Note the similarity with ax-gen 1859, ax-4 1874 and hba1 2286 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/.

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 32865) and Löb's theorem (bj-babylob 32866). See the comments of these theorems for details.

Syntaxcprvb 32859 Syntax for the provability predicate.
wff Prv 𝜑

Axiomax-prv1 32860 First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
𝜑       Prv 𝜑

Axiomax-prv2 32861 Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv (𝜑𝜓) → (Prv 𝜑 → Prv 𝜓))

Axiomax-prv3 32862 Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(Prv 𝜑 → Prv Prv 𝜑)

Theoremprvlem1 32863 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑𝜓)       (Prv 𝜑 → Prv 𝜓)

Theoremprvlem2 32864 An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.)
(𝜑 → (𝜓𝜒))       (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))

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 32866 See the section header comments for the context, as well as the comments for bj-babygodel 32865.

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/).

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

(𝜓 ↔ (Prv 𝜓𝜑))    &   (Prv 𝜑𝜑)       𝜑

Theorembj-godellob 32867 Proof of Gödel's theorem from Löb's theorem (see comments at bj-babygodel 32865 and bj-babylob 32866 for details). (Contributed by BJ, 20-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ ¬ Prv 𝜑)    &    ¬ Prv ⊥

20.14.4  First-order logic

Utility lemmas or strengthenings of theorems in the main part (biconditional or closed forms, or fewer dv conditions, or dv conditions replaced with non-freeness hypotheses...). Sorted in the same order as in the main part.

Theorembj-genr 32868 Generalization rule on the right conjunct. See 19.28 2231. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (𝜑 ∧ ∀𝑥𝜓)

Theorembj-genl 32869 Generalization rule on the left conjunct. See 19.27 2230. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorembj-genan 32870 Generalization rule on a conjunction. Forward inference associated with 19.26 1935. (Contributed by BJ, 7-Jul-2021.)
(𝜑𝜓)       (∀𝑥𝜑 ∧ ∀𝑥𝜓)

Theorembj-2alim 32871 Closed form of 2alimi 1877. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓))

Theorembj-2exim 32872 Closed form of 2eximi 1900. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓))

Theorembj-alanim 32873 Closed form of alanimi 1881. (Contributed by BJ, 6-May-2019.)
(∀𝑥((𝜑𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))

Theorembj-2albi 32874 Closed form of 2albii 1885. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓))

Theorembj-notalbii 32875 Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4082 (103>94), ballotlem2 30830 (2655>2648), bnj1143 31139 (522>519), hausdiag 21621 (2119>2104). (Contributed by BJ, 17-Jul-2021.)
(𝜑𝜓)       (∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ 𝜓)

Theorembj-2exbi 32876 Closed form of 2exbii 1912. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓))

Theorembj-3exbi 32877 Closed form of 3exbii 1913. (Contributed by BJ, 6-May-2019.)
(∀𝑥𝑦𝑧(𝜑𝜓) → (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓))

Theorembj-sylgt2 32878 Uncurried (imported) form of sylgt 1886. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜓𝜒) ∧ (𝜑 → ∀𝑥𝜓)) → (𝜑 → ∀𝑥𝜒))

Theorembj-exlimh 32879 Closed form of close to exlimih 2283. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜑𝜓) → ((∃𝑥𝜓𝜒) → (∃𝑥𝜑𝜒)))

Theorembj-exlimh2 32880 Uncurried (imported) form of bj-exlimh 32879. (Contributed by BJ, 2-May-2019.)
((∀𝑥(𝜑𝜓) ∧ (∃𝑥𝜓𝜒)) → (∃𝑥𝜑𝜒))

Theorembj-alrimhi 32881 An inference associated with sylgt 1886 and bj-exlimh 32879. (Contributed by BJ, 12-May-2019.)
(𝜑𝜓)       (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜓))

Theorembj-alexim 32882 Closed form of aleximi 1896 (with a shorter proof, so aleximi 1896 could be deduced from it -- exim 1898 would have to be proved first, but its proof is shorter (currently almost a subproof of aleximi 1896)). (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)))

Theorembj-nexdh 32883 Closed form of nexdh 1929 (actually, its general instance). (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑 → ¬ 𝜓) → ((𝜒 → ∀𝑥𝜑) → (𝜒 → ¬ ∃𝑥𝜓)))

Theorembj-nexdh2 32884 Uncurried (imported) form of bj-nexdh 32883. (Contributed by BJ, 6-May-2019.)
((∀𝑥(𝜑 → ¬ 𝜓) ∧ (𝜒 → ∀𝑥𝜑)) → (𝜒 → ¬ ∃𝑥𝜓))

Theorembj-hbxfrbi 32885 Closed form of hbxfrbi 1889. Notes: it is less important than nfbiit 1914; it requires sp 2188 (unlike nfbiit 1914); there is an obvious version with (∃𝑥𝜑𝜑) instead. (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓)))

Theorembj-exlime 32886 Variant of exlimih 2283 where the non-freeness of 𝑥 in 𝜓 is expressed using an existential quantifier, thus requiring fewer axioms. (Contributed by BJ, 17-Mar-2020.)
(∃𝑥𝜓𝜓)    &   (𝜑𝜓)       (∃𝑥𝜑𝜓)

Theorembj-exnalimn 32887 A transformation of quantifiers and logical connectives. The general statement that equs3 2029 proves.

This and the following theorems are the general instances of already proved theorems. They could be moved to the main part, before ax-5 1976. I propose to move to the main part: bj-exnalimn 32887, bj-exalim 32888, bj-exalimi 32889, bj-exalims 32890, bj-exalimsi 32891, bj-ax12i 32893, bj-ax12wlem 32894, bj-ax12w 32942, and remove equs3 2029. A new label is needed for bj-ax12i 32893 and label suggestions are welcome for the others. I also propose to change ¬ ∀𝑥¬ to 𝑥 in speimfw 2030 and spimfw 2032 (other spim* theorems use 𝑥 and very few theorems in set.mm use ¬ ∀𝑥¬). (Contributed by BJ, 29-Sep-2019.)

(∃𝑥(𝜑𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))

Theorembj-exalim 32888 Distributing quantifiers over a double implication. (Contributed by BJ, 8-Nov-2021.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒)))

Theorembj-exalimi 32889 An inference for distributing quantifiers over a double implication. (Almost) the general statement that speimfw 2030 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))

Theorembj-exalims 32890 Distributing quantifiers over a double implication. (Almost) the general statement that spimfw 2032 proves. (Contributed by BJ, 29-Sep-2019.)
(∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∀𝑥(𝜑 → (𝜓𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓𝜒)))

Theorembj-exalimsi 32891 An inference for distributing quantifiers over a double implication. (Almost) the general statement that spimfw 2032 proves. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (∃𝑥𝜑 → (¬ 𝜒 → ∀𝑥 ¬ 𝜒))       (∃𝑥𝜑 → (∀𝑥𝜓𝜒))

Theorembj-ax12ig 32892 A lemma used to prove a weak form of the axiom of substitution. A generalization of bj-ax12i 32893. (Contributed by BJ, 19-Dec-2020.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-ax12i 32893 A weakening of bj-ax12ig 32892 that is sufficient to prove a weak form of the axiom of substitution ax-12 2184. The general statement of which ax12i 2033 is an instance. (Contributed by BJ, 29-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   (𝜒 → ∀𝑥𝜒)       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

Theorembj-ax12wlem 32894* A lemma used to prove a weak version of the axiom of substitution ax-12 2184. (Temporary comment: The general statement that ax12wlem 2146 proves.) (Contributed by BJ, 20-Mar-2020.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥(𝜑𝜓)))

20.14.4.4  Equality and substitution

Theorembj-ssbjust 32895* Justification theorem for df-ssb 32897 from Tarski's FOL. (Contributed by BJ, 9-Nov-2021.)
(∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧𝜑)))

Syntaxwssb 32896 Syntax for the substitution of a variable for a variable in a formula. (Contributed by BJ, 22-Dec-2020.)
wff [𝑡/𝑥]b𝜑

Definitiondf-ssb 32897* Alternate definition of proper substitution. Note that the occurrences of a given variable are either all bound (𝑥, 𝑦) or all free (𝑡). Also note that the definiens uses only primitive symbols. It is obtained by applying twice Tarski's definition sb6 2554 which is valid for disjoint variables, so we introduce a dummy variable 𝑦 to isolate 𝑥 from 𝑡, as in dfsb7 2580 with respect to sb5 2555.

This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a DV condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row.

This definition uses a dummy variable, but the justification theorem, bj-ssbjust 32895, is provable from Tarski's FOL.

Once this is proved, more of the fundamental properties of proper substitution will be provable from Tarski's FOL system, sometimes with the help of specialization sp 2188, of the substitution axiom ax-12 2184, and of commutation of quantifiers ax-11 2171; that is, ax-13 2379 will often be avoided. (Contributed by BJ, 22-Dec-2020.)

([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))

Theorembj-ssbim 32898 Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2519. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))

Theorembj-ssbbi 32899 Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2527. (Contributed by BJ, 22-Dec-2020.)
(∀𝑥(𝜑𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓))

Theorembj-ssbimi 32900 Distribute substitution over implication. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.)
(𝜑𝜓)       ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓)

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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 43001-43033
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