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Type | Label | Description |
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Statement | ||
Theorem | knoppndvlem14 35401* | Lemma for knoppndv 35410. (Contributed by Asger C. Ipsen, 1-Jul-2021.) (Revised by Asger C. Ipsen, 7-Jul-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ πΉ = (π¦ β β β¦ (π β β0 β¦ ((πΆβπ) Β· (πβ(((2 Β· π)βπ) Β· π¦))))) & β’ π΄ = ((((2 Β· π)β-π½) / 2) Β· π) & β’ π΅ = ((((2 Β· π)β-π½) / 2) Β· (π + 1)) & β’ (π β πΆ β (-1(,)1)) & β’ (π β π½ β β0) & β’ (π β π β β€) & β’ (π β π β β) & β’ (π β 1 < (π Β· (absβπΆ))) β β’ (π β (absβ(Ξ£π β (0...(π½ β 1))((πΉβπ΅)βπ) β Ξ£π β (0...(π½ β 1))((πΉβπ΄)βπ))) β€ ((((absβπΆ)βπ½) / 2) Β· (1 / (((2 Β· π) Β· (absβπΆ)) β 1)))) | ||
Theorem | knoppndvlem15 35402* | Lemma for knoppndv 35410. (Contributed by Asger C. Ipsen, 6-Jul-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ πΉ = (π¦ β β β¦ (π β β0 β¦ ((πΆβπ) Β· (πβ(((2 Β· π)βπ) Β· π¦))))) & β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) & β’ π΄ = ((((2 Β· π)β-π½) / 2) Β· π) & β’ π΅ = ((((2 Β· π)β-π½) / 2) Β· (π + 1)) & β’ (π β πΆ β (-1(,)1)) & β’ (π β π½ β β0) & β’ (π β π β β€) & β’ (π β π β β) & β’ (π β 1 < (π Β· (absβπΆ))) β β’ (π β ((((absβπΆ)βπ½) / 2) Β· (1 β (1 / (((2 Β· π) Β· (absβπΆ)) β 1)))) β€ (absβ((πβπ΅) β (πβπ΄)))) | ||
Theorem | knoppndvlem16 35403 | Lemma for knoppndv 35410. (Contributed by Asger C. Ipsen, 19-Jul-2021.) |
β’ π΄ = ((((2 Β· π)β-π½) / 2) Β· π) & β’ π΅ = ((((2 Β· π)β-π½) / 2) Β· (π + 1)) & β’ (π β π½ β β0) & β’ (π β π β β€) & β’ (π β π β β) β β’ (π β (π΅ β π΄) = (((2 Β· π)β-π½) / 2)) | ||
Theorem | knoppndvlem17 35404* | Lemma for knoppndv 35410. (Contributed by Asger C. Ipsen, 12-Aug-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ πΉ = (π¦ β β β¦ (π β β0 β¦ ((πΆβπ) Β· (πβ(((2 Β· π)βπ) Β· π¦))))) & β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) & β’ π΄ = ((((2 Β· π)β-π½) / 2) Β· π) & β’ π΅ = ((((2 Β· π)β-π½) / 2) Β· (π + 1)) & β’ (π β πΆ β (-1(,)1)) & β’ (π β π½ β β0) & β’ (π β π β β€) & β’ (π β π β β) & β’ (π β 1 < (π Β· (absβπΆ))) β β’ (π β ((((2 Β· π) Β· (absβπΆ))βπ½) Β· (1 β (1 / (((2 Β· π) Β· (absβπΆ)) β 1)))) β€ ((absβ((πβπ΅) β (πβπ΄))) / (π΅ β π΄))) | ||
Theorem | knoppndvlem18 35405* | Lemma for knoppndv 35410. (Contributed by Asger C. Ipsen, 14-Aug-2021.) |
β’ (π β πΆ β (-1(,)1)) & β’ (π β π β β) & β’ (π β π· β β+) & β’ (π β πΈ β β+) & β’ (π β πΊ β β+) & β’ (π β 1 < (π Β· (absβπΆ))) β β’ (π β βπ β β0 ((((2 Β· π)β-π) / 2) < π· β§ πΈ β€ ((((2 Β· π) Β· (absβπΆ))βπ) Β· πΊ))) | ||
Theorem | knoppndvlem19 35406* | Lemma for knoppndv 35410. (Contributed by Asger C. Ipsen, 17-Aug-2021.) |
β’ π΄ = ((((2 Β· π)β-π½) / 2) Β· π) & β’ π΅ = ((((2 Β· π)β-π½) / 2) Β· (π + 1)) & β’ (π β π½ β β0) & β’ (π β π» β β) & β’ (π β π β β) β β’ (π β βπ β β€ (π΄ β€ π» β§ π» β€ π΅)) | ||
Theorem | knoppndvlem20 35407 | Lemma for knoppndv 35410. (Contributed by Asger C. Ipsen, 18-Aug-2021.) |
β’ (π β πΆ β (-1(,)1)) & β’ (π β π β β) & β’ (π β 1 < (π Β· (absβπΆ))) β β’ (π β (1 β (1 / (((2 Β· π) Β· (absβπΆ)) β 1))) β β+) | ||
Theorem | knoppndvlem21 35408* | Lemma for knoppndv 35410. (Contributed by Asger C. Ipsen, 18-Aug-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ πΉ = (π¦ β β β¦ (π β β0 β¦ ((πΆβπ) Β· (πβ(((2 Β· π)βπ) Β· π¦))))) & β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) & β’ πΊ = (1 β (1 / (((2 Β· π) Β· (absβπΆ)) β 1))) & β’ (π β πΆ β (-1(,)1)) & β’ (π β π· β β+) & β’ (π β πΈ β β+) & β’ (π β π» β β) & β’ (π β π½ β β0) & β’ (π β π β β) & β’ (π β 1 < (π Β· (absβπΆ))) & β’ (π β (((2 Β· π)β-π½) / 2) < π·) & β’ (π β πΈ β€ ((((2 Β· π) Β· (absβπΆ))βπ½) Β· πΊ)) β β’ (π β βπ β β βπ β β ((π β€ π» β§ π» β€ π) β§ ((π β π) < π· β§ π β π) β§ πΈ β€ ((absβ((πβπ) β (πβπ))) / (π β π)))) | ||
Theorem | knoppndvlem22 35409* | Lemma for knoppndv 35410. (Contributed by Asger C. Ipsen, 19-Aug-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ πΉ = (π¦ β β β¦ (π β β0 β¦ ((πΆβπ) Β· (πβ(((2 Β· π)βπ) Β· π¦))))) & β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) & β’ (π β πΆ β (-1(,)1)) & β’ (π β π· β β+) & β’ (π β πΈ β β+) & β’ (π β π» β β) & β’ (π β π β β) & β’ (π β 1 < (π Β· (absβπΆ))) β β’ (π β βπ β β βπ β β ((π β€ π» β§ π» β€ π) β§ ((π β π) < π· β§ π β π) β§ πΈ β€ ((absβ((πβπ) β (πβπ))) / (π β π)))) | ||
Theorem | knoppndv 35410* | The continuous nowhere differentiable function π ( Knopp, K. (1918). Math. Z. 2, 1-26 ) is, in fact, nowhere differentiable. (Contributed by Asger C. Ipsen, 19-Aug-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ πΉ = (π¦ β β β¦ (π β β0 β¦ ((πΆβπ) Β· (πβ(((2 Β· π)βπ) Β· π¦))))) & β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) & β’ (π β πΆ β (-1(,)1)) & β’ (π β π β β) & β’ (π β 1 < (π Β· (absβπΆ))) β β’ (π β dom (β D π) = β ) | ||
Theorem | knoppf 35411* | Knopp's function is a function. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ πΉ = (π¦ β β β¦ (π β β0 β¦ ((πΆβπ) Β· (πβ(((2 Β· π)βπ) Β· π¦))))) & β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) & β’ (π β πΆ β (-1(,)1)) & β’ (π β π β β) β β’ (π β π:ββΆβ) | ||
Theorem | knoppcn2 35412* | Variant of knoppcn 35380 with different codomain. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ πΉ = (π¦ β β β¦ (π β β0 β¦ ((πΆβπ) Β· (πβ(((2 Β· π)βπ) Β· π¦))))) & β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) & β’ (π β π β β) & β’ (π β πΆ β (-1(,)1)) β β’ (π β π β (ββcnββ)) | ||
Theorem | cnndvlem1 35413* | Lemma for cnndv 35415. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ πΉ = (π¦ β β β¦ (π β β0 β¦ (((1 / 2)βπ) Β· (πβ(((2 Β· 3)βπ) Β· π¦))))) & β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) β β’ (π β (ββcnββ) β§ dom (β D π) = β ) | ||
Theorem | cnndvlem2 35414* | Lemma for cnndv 35415. (Contributed by Asger C. Ipsen, 26-Aug-2021.) |
β’ π = (π₯ β β β¦ (absβ((ββ(π₯ + (1 / 2))) β π₯))) & β’ πΉ = (π¦ β β β¦ (π β β0 β¦ (((1 / 2)βπ) Β· (πβ(((2 Β· 3)βπ) Β· π¦))))) & β’ π = (π€ β β β¦ Ξ£π β β0 ((πΉβπ€)βπ)) β β’ βπ(π β (ββcnββ) β§ dom (β D π) = β ) | ||
Theorem | cnndv 35415 | There exists a continuous nowhere differentiable function. The result follows directly from knoppcn 35380 and knoppndv 35410. (Contributed by Asger C. Ipsen, 26-Aug-2021.) |
β’ βπ(π β (ββcnββ) β§ dom (β D π) = β ) | ||
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. | ||
Miscellaneous utility theorems of propositional calculus. | ||
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. | ||
Theorem | bj-mp2c 35416 | A double modus ponens inference. Inference associated with mpd 15. (Contributed by BJ, 24-Sep-2019.) |
β’ π & β’ (π β π) & β’ (π β (π β π)) β β’ π | ||
Theorem | bj-mp2d 35417 | A double modus ponens inference. Inference associated with mpcom 38. (Contributed by BJ, 24-Sep-2019.) |
β’ π & β’ (π β π) & β’ (π β (π β π)) β β’ π | ||
In this section, we prove a syntactic theorem (bj-0 35418) asserting that some formula is well-formed. Then, we use this syntactic theorem to shorten the proof of a "usual" theorem (bj-1 35419) and explain in the comment of that theorem why this phenomenon is unusual. | ||
Theorem | bj-0 35418 | A syntactic theorem. See the section comment and the comment of bj-1 35419. 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 2108 and weq 1967. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
wff ((π β π) β π) | ||
Theorem | bj-1 35419 |
In this proof, the use of the syntactic theorem bj-0 35418
allows to reduce
the total length by one (non-essential) step. See also the section
comment and the comment of bj-0 35418. Since bj-0 35418
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 35418
as a theorem referencing
it). The full proof reads $= wph wps wch bj-0 id $. (while, without
using bj-0 35418, 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 1967 or wel 2108). 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 35419 is a special instance of id 22. (Contributed by BJ, 24-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (((π β π) β π) β ((π β π) β π)) | ||
Theorem | bj-a1k 35420 | 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 35420 does not require ax-2 7). This shortens the proofs of dfwe2 7761 (937>925), ordunisuc2 7833 (789>777), r111 9770 (558>545), smo11 8364 (1176>1164). (Contributed by BJ, 11-Aug-2020.) (Proof modification is discouraged.) |
β’ (π β (π β (π β π))) | ||
Theorem | bj-poni 35421 | Inference associated with "pon", pm2.27 42. Its associated inference is ax-mp 5. (Contributed by BJ, 30-Jul-2024.) |
β’ π β β’ ((π β π) β π) | ||
Theorem | bj-nnclav 35422 | When β₯ is substituted for π, 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.) |
β’ (((π β π) β π) β ((π β π) β π)) | ||
Theorem | bj-nnclavi 35423 | Inference associated with bj-nnclav 35422. Its associated inference is an instance of syl 17. Notice the non-intuitionistic proof from bj-peircei 35442 and bj-poni 35421. (Contributed by BJ, 30-Jul-2024.) |
β’ ((π β π) β π) β β’ ((π β π) β π) | ||
Theorem | bj-nnclavc 35424 | Commuted form of bj-nnclav 35422. Notice the non-intuitionistic proof from bj-peircei 35442 and imim1i 63. (Contributed by BJ, 30-Jul-2024.) A proof which is shorter when compressed uses embantd 59. (Proof modification is discouraged.) |
β’ ((π β π) β (((π β π) β π) β π)) | ||
Theorem | bj-nnclavci 35425 | Inference associated with bj-nnclavc 35424. 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.) |
β’ (π β π) β β’ (((π β π) β π) β π) | ||
Theorem | bj-jarrii 35426 | 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.) |
β’ ((π β π) β π) & β’ π β β’ π | ||
Theorem | bj-imim21 35427 | The propositional function (π β (. β π)) is decreasing. (Contributed by BJ, 19-Jul-2019.) |
β’ ((π β π) β ((π β (π β π)) β (π β (π β π)))) | ||
Theorem | bj-imim21i 35428 | Inference associated with bj-imim21 35427. Its associated inference is syl5 34. (Contributed by BJ, 19-Jul-2019.) |
β’ (π β π) β β’ ((π β (π β π)) β (π β (π β π))) | ||
Theorem | bj-peircestab 35429 | Over minimal implicational calculus, Peirce's law implies the double negation of the stability of any formula (that is the interpretation when β₯ is substituted for π and for π). 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 35430). (Contributed by BJ, 30-Nov-2023.) Axiom ax-3 8 is only used through Peirce's law peirce 201. (Proof modification is discouraged.) |
β’ (((((π β π) β π) β π) β π) β π) | ||
Theorem | bj-stabpeirce 35430 | This minimal implicational calculus tautology is used in the following argument: When π, π, π, π, π are replaced respectively by (π β β₯), β₯, π, β₯, β₯, the antecedent becomes Β¬ Β¬ (Β¬ Β¬ π β π), that is, the double negation of the stability of π. If that statement were provable in minimal calculus, then, since β₯ plays no particular role in minimal calculus, also the statement with π in place of β₯ would be provable. The corresponding consequent is (((π β π) β π) β π), 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 35429). (Contributed by BJ, 30-Nov-2023.) (Revised by BJ, 30-Jul-2024.) (Proof modification is discouraged.) |
β’ (((((π β π) β π) β π) β π) β (((π β π) β π) β π)) | ||
Positive calculus is understood to be intuitionistic. | ||
Theorem | bj-syl66ib 35431 | A mixed syllogism inference derived from imbitrdi 250. In addition to bj-dvelimdv1 35731, it can also shorten alexsubALTlem4 23554 (4821>4812), supsrlem 11106 (2868>2863). (Contributed by BJ, 20-Oct-2021.) |
β’ (π β (π β π)) & β’ (π β π) & β’ (π β π) β β’ (π β (π β π)) | ||
Theorem | bj-orim2 35432 | Proof of orim2 967 from the axiomatic definition of disjunction (olc 867, orc 866, jao 960) and minimal implicational calculus. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) |
β’ ((π β π) β ((π β¨ π) β (π β¨ π))) | ||
Theorem | bj-currypeirce 35433 | Curry's axiom curryax 893 (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 960 via its inference form jaoi 856; the introduction axioms olc 867 and orc 866 are not needed). Note that this theorem shows that actually, the standard instance of curryax 893 implies the standard instance of peirce 201, which is not the case for the converse bj-peircecurry 35434. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((π β¨ (π β π)) β (((π β π) β π) β π)) | ||
Theorem | bj-peircecurry 35434 | Peirce's axiom peirce 201 implies Curry's axiom curryax 893 over minimal implicational calculus and the axiomatic definition of disjunction (actually, only the introduction axioms olc 867 and orc 866; the elimination axiom jao 960 is not needed). See bj-currypeirce 35433 for the converse. (Contributed by BJ, 15-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β¨ (π β π)) | ||
Theorem | bj-animbi 35435 | 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.) |
β’ ((π β§ π) β (π β (π β π))) | ||
Theorem | bj-currypara 35436 | 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 π is the self-referential sentence "If this sentence is true, then π", 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.) |
β’ ((π β (π β π)) β π) | ||
Theorem | bj-con2com 35437 | A commuted form of the contrapositive, true in minimal calculus. (Contributed by BJ, 19-Mar-2020.) |
β’ (π β ((π β Β¬ π) β Β¬ π)) | ||
Theorem | bj-con2comi 35438 | Inference associated with bj-con2com 35437. 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 35438. (Contributed by BJ, 19-Mar-2020.) |
β’ π β β’ ((π β Β¬ π) β Β¬ π) | ||
Theorem | bj-pm2.01i 35439 | Inference associated with the weak Clavius law pm2.01 188. (Contributed by BJ, 30-Mar-2020.) |
β’ (π β Β¬ π) β β’ Β¬ π | ||
Theorem | bj-nimn 35440 | 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.) |
β’ (π β Β¬ (π β Β¬ π)) | ||
Theorem | bj-nimni 35441 | Inference associated with bj-nimn 35440. (Contributed by BJ, 19-Mar-2020.) |
β’ π β β’ Β¬ (π β Β¬ π) | ||
Theorem | bj-peircei 35442 | Inference associated with peirce 201. (Contributed by BJ, 30-Mar-2020.) |
β’ ((π β π) β π) β β’ π | ||
Theorem | bj-looinvi 35443 | Inference associated with looinv 202. Its associated inference is bj-looinvii 35444. (Contributed by BJ, 30-Mar-2020.) |
β’ ((π β π) β π) β β’ ((π β π) β π) | ||
Theorem | bj-looinvii 35444 | Inference associated with bj-looinvi 35443. (Contributed by BJ, 30-Mar-2020.) |
β’ ((π β π) β π) & β’ (π β π) β β’ π | ||
Theorem | bj-mt2bi 35445 | Version of mt2 199 where the major premise is a biconditional. Another proof is also possible via con2bii 358 and mpbi 229. The current mt2bi 364 should be relabeled, maybe to imfal. (Contributed by BJ, 5-Oct-2024.) |
β’ π & β’ (π β Β¬ π) β β’ Β¬ π | ||
Theorem | bj-ntrufal 35446 | The negation of a theorem is equivalent to false. This can shorten dfnul2 4326. (Contributed by BJ, 5-Oct-2024.) |
β’ π β β’ (Β¬ π β β₯) | ||
Theorem | bj-fal 35447 | Shortening of fal 1556 using bj-mt2bi 35445. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) (Proof modification is discouraged.) |
β’ Β¬ β₯ | ||
A few lemmas about disjunction. The fundamental theorems in this family are the dual statements pm4.71 559 and pm4.72 949. See also biort 935 and biorf 936. | ||
Theorem | bj-jaoi1 35448 | Shortens orfa2 36954 (58>53), pm1.2 903 (20>18), pm1.2 903 (20>18), pm2.4 906 (31>25), pm2.41 907 (31>25), pm2.42 942 (38>32), pm3.2ni 880 (43>39), pm4.44 996 (55>51). (Contributed by BJ, 30-Sep-2019.) |
β’ (π β π) β β’ ((π β¨ π) β π) | ||
Theorem | bj-jaoi2 35449 | Shortens consensus 1052 (110>106), elnn0z 12571 (336>329), pm1.2 903 (20>19), pm3.2ni 880 (43>39), pm4.44 996 (55>51). (Contributed by BJ, 30-Sep-2019.) |
β’ (π β π) β β’ ((π β¨ π) β π) | ||
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 847, df-an 398, pm4.64 848, imor 852, pm4.62 855 through pm4.67 400, and, for the De Morgan laws, ianor 981 through pm4.57 990. | ||
Theorem | bj-dfbi4 35450 | Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
β’ ((π β π) β ((π β§ π) β¨ Β¬ (π β¨ π))) | ||
Theorem | bj-dfbi5 35451 | Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
β’ ((π β π) β ((π β¨ π) β (π β§ π))) | ||
Theorem | bj-dfbi6 35452 | Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
β’ ((π β π) β ((π β¨ π) β (π β§ π))) | ||
Theorem | bj-bijust0ALT 35453 | 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.) |
β’ Β¬ ((π β π) β Β¬ (π β π)) | ||
Theorem | bj-bijust00 35454 | 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 35453 are therefore also instances of it). (Contributed by BJ, 7-Sep-2022.) |
β’ Β¬ ((π β π) β Β¬ (π β π)) | ||
Theorem | bj-consensus 35455 | Version of consensus 1052 expressed using the conditional operator. (Remark: it may be better to express it as consensus 1052, 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.) |
β’ ((if-(π, π, π) β¨ (π β§ π)) β if-(π, π, π)) | ||
Theorem | bj-consensusALT 35456 | Alternate proof of bj-consensus 35455. (Contributed by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((if-(π, π, π) β¨ (π β§ π)) β if-(π, π, π)) | ||
Theorem | bj-df-ifc 35457* | 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 2711. We reprove the current df-if 4530 from it in bj-dfif 35458. (Contributed by BJ, 20-Sep-2019.) (Proof modification is discouraged.) |
β’ if(π, π΄, π΅) = {π₯ β£ if-(π, π₯ β π΄, π₯ β π΅)} | ||
Theorem | bj-dfif 35458* | 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.) |
β’ if(π, π΄, π΅) = {π₯ β£ ((π β§ π₯ β π΄) β¨ (Β¬ π β§ π₯ β π΅))} | ||
Theorem | bj-ififc 35459 | A biconditional connecting the conditional operator for propositions and the conditional operator for classes. Note that there is no sethood hypothesis on π: it is implied by either side. (Contributed by BJ, 24-Sep-2019.) Generalize statement from setvar π₯ to class π. (Revised by BJ, 26-Dec-2023.) |
β’ (π β if(π, π΄, π΅) β if-(π, π β π΄, π β π΅)) | ||
Miscellaneous theorems of propositional calculus. | ||
Theorem | bj-imbi12 35460 | Uncurried (imported) form of imbi12 347. (Contributed by BJ, 6-May-2019.) |
β’ (((π β π) β§ (π β π)) β ((π β π) β (π β π))) | ||
Theorem | bj-biorfi 35461 | This should be labeled "biorfi" while the current biorfi 938 should be labeled "biorfri". The dual of biorf 936 is not biantr 805 but iba 529 (and ibar 530). 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.) |
β’ Β¬ π β β’ (π β (π β¨ π)) | ||
Theorem | bj-falor 35462 | Dual of truan 1553 (which has biconditional reversed). (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.) |
β’ (π β (β₯ β¨ π)) | ||
Theorem | bj-falor2 35463 | Dual of truan 1553. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.) |
β’ ((β₯ β¨ π) β π) | ||
Theorem | bj-bibibi 35464 | A property of the biconditional. (Contributed by BJ, 26-Oct-2019.) (Proof modification is discouraged.) |
β’ (π β (π β (π β π))) | ||
Theorem | bj-imn3ani 35465 | Duplication of bnj1224 33812. Three-fold version of imnani 402. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by BJ, 22-Oct-2019.) (Proof modification is discouraged.) |
β’ Β¬ (π β§ π β§ π) β β’ ((π β§ π) β Β¬ π) | ||
Theorem | bj-andnotim 35466 | 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.) |
β’ (((π β§ Β¬ π) β π) β ((π β π) β¨ π)) | ||
Theorem | bj-bi3ant 35467 | This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.) |
β’ (π β (π β π)) β β’ (((π β π) β π) β (((π β π) β π) β ((π β π) β π))) | ||
Theorem | bj-bisym 35468 | This used to be in the main part. (Contributed by Wolf Lammen, 14-May-2013.) (Revised by BJ, 14-Jun-2019.) |
β’ (((π β π) β (π β π)) β (((π β π) β (π β π)) β ((π β π) β (π β π)))) | ||
Theorem | bj-bixor 35469 | Equivalence of two ternary operations. Note the identical order and parenthesizing of the three arguments in both expressions. (Contributed by BJ, 31-Dec-2023.) |
β’ ((π β (π β» π)) β (π β» (π β π))) | ||
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 βπ₯ to "necessity" (generally denoted by a box) and βπ₯ 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 π₯ and any other metavariables appearing in the statements.) For instance, ax-gen 1798 corresponds to the necessitation rule of modal logic, and ax-4 1812 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 GΓΆdelβLΓΆb 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/ 1812. A basic result in this logic is bj-gl4 35473. | ||
Theorem | bj-axdd2 35470 | This implication, proved using only ax-gen 1798 and ax-4 1812 on top of propositional calculus (hence holding, up to the standard interpretation, in any normal modal logic), shows that the axiom scheme β’ βπ₯β€ implies the axiom scheme β’ (βπ₯π β βπ₯π). 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 35471. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (βπ₯π β (βπ₯π β βπ₯π)) | ||
Theorem | bj-axd2d 35471 | This implication, proved using only ax-gen 1798 on top of propositional calculus (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme β’ (βπ₯π β βπ₯π) implies the axiom scheme β’ βπ₯β€. 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 35470. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((βπ₯β€ β βπ₯β€) β βπ₯β€) | ||
Theorem | bj-axtd 35472 | This implication, proved from propositional calculus only (hence holding, up to the standard interpretation, in any modal logic), shows that the axiom scheme β’ (βπ₯π β π) (modal T) implies the axiom scheme β’ (βπ₯π β βπ₯π) (modal D). See also bj-axdd2 35470 and bj-axd2d 35471. (Contributed by BJ, 16-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((βπ₯ Β¬ π β Β¬ π) β ((βπ₯π β π) β (βπ₯π β βπ₯π))) | ||
Theorem | bj-gl4 35473 | In a normal modal logic, the modal axiom GL implies the modal axiom (4). Translated to first-order logic, Axiom GL reads β’ (βπ₯(βπ₯π β π) β βπ₯π). Note that the antecedent of bj-gl4 35473 is an instance of the axiom GL, with π replaced by (βπ₯π β§ π), which is a modality sometimes called the "strong necessity" of π. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((βπ₯(βπ₯(βπ₯π β§ π) β (βπ₯π β§ π)) β βπ₯(βπ₯π β§ π)) β (βπ₯π β βπ₯βπ₯π)) | ||
Theorem | bj-axc4 35474 | Over minimal calculus, the modal axiom (4) (hba1 2290) and the modal axiom (K) (ax-4 1812) together imply axc4 2315. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ ((βπ₯π β βπ₯βπ₯π) β ((βπ₯(βπ₯π β π) β (βπ₯βπ₯π β βπ₯π)) β (βπ₯(βπ₯π β π) β (βπ₯π β βπ₯π)))) | ||
In this section, we assume that, on top of propositional calculus, there is given a provability predicate Prv satisfying the three axioms ax-prv1 35476 and ax-prv2 35477 and ax-prv3 35478. Note the similarity with ax-gen 1798, ax-4 1812 and hba1 2290 respectively. These three properties of Prv are often called the HilbertβBernaysβLΓΆb derivability conditions, or the HilbertβBernays 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 π means "π is provable in T", and the turnstile β’ indicates provability in T. Beware that "provability logic" often means (K) augmented with the GΓΆdelβLΓΆb axiom GL, which we do not assume here (at least for the moment). See for instance https://plato.stanford.edu/entries/logic-provability/ 2290. 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 GΓΆdel 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 GΓΆdel's second incompleteness theorem (bj-babygodel 35481) and LΓΆb's theorem (bj-babylob 35482). See the comments of these theorems for details. | ||
Syntax | cprvb 35475 | Syntax for the provability predicate. |
wff Prv π | ||
Axiom | ax-prv1 35476 | First property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
β’ π β β’ Prv π | ||
Axiom | ax-prv2 35477 | Second property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
β’ (Prv (π β π) β (Prv π β Prv π)) | ||
Axiom | ax-prv3 35478 | Third property of three of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
β’ (Prv π β Prv Prv π) | ||
Theorem | prvlem1 35479 | An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
β’ (π β π) β β’ (Prv π β Prv π) | ||
Theorem | prvlem2 35480 | An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019.) |
β’ (π β (π β π)) β β’ (Prv π β (Prv π β Prv π)) | ||
Theorem | bj-babygodel 35481 |
See the section header comments for the context.
The first hypothesis reads "π 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 π is a formal version of the sentence "This sentence is not provable". The hard part of the proof of GΓΆdel's theorem is to construct such a π, called a "GΓΆdelβRosser sentence", for a first-order theory T which is effectively axiomatizable and contains Robinson arithmetic, through GΓΆdel diagonalization (this can be done in primitive recursive arithmetic). The second hypothesis means that β₯ 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, GΓΆdel'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.) |
β’ (π β Β¬ Prv π) & β’ Β¬ Prv β₯ β β’ β₯ | ||
Theorem | bj-babylob 35482 |
See the section header comments for the context, as well as the comments
for bj-babygodel 35481.
LΓΆb's theorem when the LΓΆb sentence is given as a hypothesis (the hard part of the proof of LΓΆb's theorem is to construct this LΓΆb sentence; this can be done, using GΓΆdel 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 LΓΆb sentence, given here as the first hypothesis), then the theory actually proves that sentence. See for instance, Eliezer Yudkowsky, The Cartoon Guide to LΓΆb's Theorem (available at http://yudkowsky.net/rational/lobs-theorem/ 35481). (Contributed by BJ, 20-Apr-2019.) |
β’ (π β (Prv π β π)) & β’ (Prv π β π) β β’ π | ||
Theorem | bj-godellob 35483 | Proof of GΓΆdel's theorem from LΓΆb's theorem (see comments at bj-babygodel 35481 and bj-babylob 35482 for details). (Contributed by BJ, 20-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π β Β¬ Prv π) & β’ Β¬ Prv β₯ β β’ β₯ | ||
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. | ||
Theorem | bj-genr 35484 | Generalization rule on the right conjunct. See 19.28 2222. (Contributed by BJ, 7-Jul-2021.) |
β’ (π β§ π) β β’ (π β§ βπ₯π) | ||
Theorem | bj-genl 35485 | Generalization rule on the left conjunct. See 19.27 2221. (Contributed by BJ, 7-Jul-2021.) |
β’ (π β§ π) β β’ (βπ₯π β§ π) | ||
Theorem | bj-genan 35486 | Generalization rule on a conjunction. Forward inference associated with 19.26 1874. (Contributed by BJ, 7-Jul-2021.) |
β’ (π β§ π) β β’ (βπ₯π β§ βπ₯π) | ||
Theorem | bj-mpgs 35487 | 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 β’ π β β’ π. Strong necessity is stronger than necessity, and equivalent to it when sp 2177 (modal T) is available. Therefore, this theorem is stronger than mpg 1800 when sp 2177 is not available. (Contributed by BJ, 1-Nov-2023.) |
β’ π & β’ ((π β§ βπ₯π) β π) β β’ π | ||
Theorem | bj-2alim 35488 | Closed form of 2alimi 1815. (Contributed by BJ, 6-May-2019.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | bj-2exim 35489 | Closed form of 2eximi 1839. (Contributed by BJ, 6-May-2019.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | bj-alanim 35490 | Closed form of alanimi 1819. (Contributed by BJ, 6-May-2019.) |
β’ (βπ₯((π β§ π) β π) β ((βπ₯π β§ βπ₯π) β βπ₯π)) | ||
Theorem | bj-2albi 35491 | Closed form of 2albii 1823. (Contributed by BJ, 6-May-2019.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | bj-notalbii 35492 | Equivalence of universal quantification of negation of equivalent formulas. Shortens ab0 4375 (103>94), ballotlem2 33487 (2655>2648), bnj1143 33801 (522>519), hausdiag 23149 (2119>2104). (Contributed by BJ, 17-Jul-2021.) |
β’ (π β π) β β’ (βπ₯ Β¬ π β βπ₯ Β¬ π) | ||
Theorem | bj-2exbi 35493 | Closed form of 2exbii 1852. (Contributed by BJ, 6-May-2019.) |
β’ (βπ₯βπ¦(π β π) β (βπ₯βπ¦π β βπ₯βπ¦π)) | ||
Theorem | bj-3exbi 35494 | Closed form of 3exbii 1853. (Contributed by BJ, 6-May-2019.) |
β’ (βπ₯βπ¦βπ§(π β π) β (βπ₯βπ¦βπ§π β βπ₯βπ¦βπ§π)) | ||
Theorem | bj-sylgt2 35495 | Uncurried (imported) form of sylgt 1825. (Contributed by BJ, 2-May-2019.) |
β’ ((βπ₯(π β π) β§ (π β βπ₯π)) β (π β βπ₯π)) | ||
Theorem | bj-alrimg 35496 | The general form of the *alrim* family of theorems: if π is substituted for π, then the antecedent expresses a form of nonfreeness of π₯ in π, 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 35500. (Contributed by BJ, 9-Dec-2023.) |
β’ ((π β βπ₯π) β (βπ₯(π β π) β (π β βπ₯π))) | ||
Theorem | bj-alrimd 35497 | A slightly more general alrimd 2209. A common usage will have π substituted for π and π substituted for π, giving a form closer to alrimd 2209. (Contributed by BJ, 25-Dec-2023.) |
β’ (π β βπ₯π) & β’ (π β (π β βπ₯π)) & β’ (π β (π β π)) β β’ (π β (π β βπ₯π)) | ||
Theorem | bj-sylget 35498 | Dual statement of sylgt 1825. Closed form of bj-sylge 35501. (Contributed by BJ, 2-May-2019.) |
β’ (βπ₯(π β π) β ((βπ₯π β π) β (βπ₯π β π))) | ||
Theorem | bj-sylget2 35499 | Uncurried (imported) form of bj-sylget 35498. (Contributed by BJ, 2-May-2019.) |
β’ ((βπ₯(π β π) β§ (βπ₯π β π)) β (βπ₯π β π)) | ||
Theorem | bj-exlimg 35500 | The general form of the *exlim* family of theorems: if π is substituted for π, then the antecedent expresses a form of nonfreeness of π₯ in π, 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 35496. (Contributed by BJ, 9-Dec-2023.) |
β’ ((βπ₯π β π) β (βπ₯(π β π) β (βπ₯π β π))) |
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