![]() |
Metamath
Proof Explorer Theorem List (p. 439 of 482) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30715) |
![]() (30716-32238) |
![]() (32239-48161) |
Type | Label | Description | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Statement | ||||||||||||||||||||||||||||||||
Theorem | compne 43801 | The complement of ๐ด is not equal to ๐ด. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.) | ||||||||||||||||||||||||||||||
โข (V โ ๐ด) โ ๐ด | ||||||||||||||||||||||||||||||||
Theorem | compab 43802 | Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) | ||||||||||||||||||||||||||||||
โข (V โ {๐ง โฃ ๐}) = {๐ง โฃ ยฌ ๐} | ||||||||||||||||||||||||||||||||
Theorem | conss2 43803 | Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) | ||||||||||||||||||||||||||||||
โข (๐ด โ (V โ ๐ต) โ ๐ต โ (V โ ๐ด)) | ||||||||||||||||||||||||||||||||
Theorem | conss1 43804 | Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) | ||||||||||||||||||||||||||||||
โข ((V โ ๐ด) โ ๐ต โ (V โ ๐ต) โ ๐ด) | ||||||||||||||||||||||||||||||||
Theorem | ralbidar 43805 | More general form of ralbida 3262. (Contributed by Andrew Salmon, 25-Jul-2011.) | ||||||||||||||||||||||||||||||
โข (๐ โ โ๐ฅ โ ๐ด ๐) & โข ((๐ โง ๐ฅ โ ๐ด) โ (๐ โ ๐)) โ โข (๐ โ (โ๐ฅ โ ๐ด ๐ โ โ๐ฅ โ ๐ด ๐)) | ||||||||||||||||||||||||||||||||
Theorem | rexbidar 43806 | More general form of rexbida 3264. (Contributed by Andrew Salmon, 25-Jul-2011.) | ||||||||||||||||||||||||||||||
โข (๐ โ โ๐ฅ โ ๐ด ๐) & โข ((๐ โง ๐ฅ โ ๐ด) โ (๐ โ ๐)) โ โข (๐ โ (โ๐ฅ โ ๐ด ๐ โ โ๐ฅ โ ๐ด ๐)) | ||||||||||||||||||||||||||||||||
Theorem | dropab1 43807 | Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.) | ||||||||||||||||||||||||||||||
โข (โ๐ฅ ๐ฅ = ๐ฆ โ {โจ๐ฅ, ๐งโฉ โฃ ๐} = {โจ๐ฆ, ๐งโฉ โฃ ๐}) | ||||||||||||||||||||||||||||||||
Theorem | dropab2 43808 | Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.) | ||||||||||||||||||||||||||||||
โข (โ๐ฅ ๐ฅ = ๐ฆ โ {โจ๐ง, ๐ฅโฉ โฃ ๐} = {โจ๐ง, ๐ฆโฉ โฃ ๐}) | ||||||||||||||||||||||||||||||||
Theorem | ipo0 43809 | If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.) | ||||||||||||||||||||||||||||||
โข ( I Po ๐ด โ ๐ด = โ ) | ||||||||||||||||||||||||||||||||
Theorem | ifr0 43810 | A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.) | ||||||||||||||||||||||||||||||
โข ( I Fr ๐ด โ ๐ด = โ ) | ||||||||||||||||||||||||||||||||
Theorem | ordpss 43811 | ordelpss 6391 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.) | ||||||||||||||||||||||||||||||
โข (Ord ๐ต โ (๐ด โ ๐ต โ ๐ด โ ๐ต)) | ||||||||||||||||||||||||||||||||
Theorem | fvsb 43812* | Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.) | ||||||||||||||||||||||||||||||
โข (โ!๐ฆ ๐ด๐น๐ฆ โ ([(๐นโ๐ด) / ๐ฅ]๐ โ โ๐ฅ(โ๐ฆ(๐ด๐น๐ฆ โ ๐ฆ = ๐ฅ) โง ๐))) | ||||||||||||||||||||||||||||||||
Theorem | fveqsb 43813* | Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.) | ||||||||||||||||||||||||||||||
โข (๐ฅ = (๐นโ๐ด) โ (๐ โ ๐)) & โข โฒ๐ฅ๐ โ โข (โ!๐ฆ ๐ด๐น๐ฆ โ (๐ โ โ๐ฅ(โ๐ฆ(๐ด๐น๐ฆ โ ๐ฆ = ๐ฅ) โง ๐))) | ||||||||||||||||||||||||||||||||
Theorem | xpexb 43814 | A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.) | ||||||||||||||||||||||||||||||
โข ((๐ด ร ๐ต) โ V โ (๐ต ร ๐ด) โ V) | ||||||||||||||||||||||||||||||||
Theorem | trelpss 43815 | An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5656, ax-reg 9607 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.) | ||||||||||||||||||||||||||||||
โข ((Tr ๐ด โง ๐ต โ ๐ด) โ ๐ต โ ๐ด) | ||||||||||||||||||||||||||||||||
Theorem | addcomgi 43816 | Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.) | ||||||||||||||||||||||||||||||
โข (๐ด + ๐ต) = (๐ต + ๐ด) | ||||||||||||||||||||||||||||||||
Syntax | cplusr 43817 | Introduce the operation of vector addition. | ||||||||||||||||||||||||||||||
class +๐ | ||||||||||||||||||||||||||||||||
Syntax | cminusr 43818 | Introduce the operation of vector subtraction. | ||||||||||||||||||||||||||||||
class -๐ | ||||||||||||||||||||||||||||||||
Syntax | ctimesr 43819 | Introduce the operation of scalar multiplication. | ||||||||||||||||||||||||||||||
class .๐ฃ | ||||||||||||||||||||||||||||||||
Syntax | cptdfc 43820 | PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems. | ||||||||||||||||||||||||||||||
class PtDf(๐ด, ๐ต) | ||||||||||||||||||||||||||||||||
Syntax | crr3c 43821 | RR3 is a class. | ||||||||||||||||||||||||||||||
class RR3 | ||||||||||||||||||||||||||||||||
Syntax | cline3 43822 | line3 is a class. | ||||||||||||||||||||||||||||||
class line3 | ||||||||||||||||||||||||||||||||
Definition | df-addr 43823* | Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข +๐ = (๐ฅ โ V, ๐ฆ โ V โฆ (๐ฃ โ โ โฆ ((๐ฅโ๐ฃ) + (๐ฆโ๐ฃ)))) | ||||||||||||||||||||||||||||||||
Definition | df-subr 43824* | Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข -๐ = (๐ฅ โ V, ๐ฆ โ V โฆ (๐ฃ โ โ โฆ ((๐ฅโ๐ฃ) โ (๐ฆโ๐ฃ)))) | ||||||||||||||||||||||||||||||||
Definition | df-mulv 43825* | Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข .๐ฃ = (๐ฅ โ V, ๐ฆ โ V โฆ (๐ฃ โ โ โฆ (๐ฅ ยท (๐ฆโ๐ฃ)))) | ||||||||||||||||||||||||||||||||
Theorem | addrval 43826* | Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ถ โง ๐ต โ ๐ท) โ (๐ด+๐๐ต) = (๐ฃ โ โ โฆ ((๐ดโ๐ฃ) + (๐ตโ๐ฃ)))) | ||||||||||||||||||||||||||||||||
Theorem | subrval 43827* | Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ถ โง ๐ต โ ๐ท) โ (๐ด-๐๐ต) = (๐ฃ โ โ โฆ ((๐ดโ๐ฃ) โ (๐ตโ๐ฃ)))) | ||||||||||||||||||||||||||||||||
Theorem | mulvval 43828* | Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ถ โง ๐ต โ ๐ท) โ (๐ด.๐ฃ๐ต) = (๐ฃ โ โ โฆ (๐ด ยท (๐ตโ๐ฃ)))) | ||||||||||||||||||||||||||||||||
Theorem | addrfv 43829 | Vector addition at a value. The operation takes each vector ๐ด and ๐ต and forms a new vector whose values are the sum of each of the values of ๐ด and ๐ต. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ธ โง ๐ต โ ๐ท โง ๐ถ โ โ) โ ((๐ด+๐๐ต)โ๐ถ) = ((๐ดโ๐ถ) + (๐ตโ๐ถ))) | ||||||||||||||||||||||||||||||||
Theorem | subrfv 43830 | Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ธ โง ๐ต โ ๐ท โง ๐ถ โ โ) โ ((๐ด-๐๐ต)โ๐ถ) = ((๐ดโ๐ถ) โ (๐ตโ๐ถ))) | ||||||||||||||||||||||||||||||||
Theorem | mulvfv 43831 | Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ธ โง ๐ต โ ๐ท โง ๐ถ โ โ) โ ((๐ด.๐ฃ๐ต)โ๐ถ) = (๐ด ยท (๐ตโ๐ถ))) | ||||||||||||||||||||||||||||||||
Theorem | addrfn 43832 | Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ถ โง ๐ต โ ๐ท) โ (๐ด+๐๐ต) Fn โ) | ||||||||||||||||||||||||||||||||
Theorem | subrfn 43833 | Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ถ โง ๐ต โ ๐ท) โ (๐ด-๐๐ต) Fn โ) | ||||||||||||||||||||||||||||||||
Theorem | mulvfn 43834 | Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ถ โง ๐ต โ ๐ท) โ (๐ด.๐ฃ๐ต) Fn โ) | ||||||||||||||||||||||||||||||||
Theorem | addrcom 43835 | Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ถ โง ๐ต โ ๐ท) โ (๐ด+๐๐ต) = (๐ต+๐๐ด)) | ||||||||||||||||||||||||||||||||
Definition | df-ptdf 43836* | Define the predicate PtDf, which is a utility definition used to shorten definitions and simplify proofs. (Contributed by Andrew Salmon, 15-Jul-2012.) | ||||||||||||||||||||||||||||||
โข PtDf(๐ด, ๐ต) = (๐ฅ โ โ โฆ (((๐ฅ.๐ฃ(๐ต-๐๐ด)) +๐ฃ ๐ด) โ {1, 2, 3})) | ||||||||||||||||||||||||||||||||
Definition | df-rr3 43837 | Define the set of all points RR3. We define each point ๐ด as a function to allow the use of vector addition and subtraction as well as scalar multiplication in our proofs. (Contributed by Andrew Salmon, 15-Jul-2012.) | ||||||||||||||||||||||||||||||
โข RR3 = (โ โm {1, 2, 3}) | ||||||||||||||||||||||||||||||||
Definition | df-line3 43838* | Define the set of all lines. A line is an infinite subset of RR3 that satisfies a PtDf property. (Contributed by Andrew Salmon, 15-Jul-2012.) | ||||||||||||||||||||||||||||||
โข line3 = {๐ฅ โ ๐ซ RR3 โฃ (2o โผ ๐ฅ โง โ๐ฆ โ ๐ฅ โ๐ง โ ๐ฅ (๐ง โ ๐ฆ โ ran PtDf(๐ฆ, ๐ง) = ๐ฅ))} | ||||||||||||||||||||||||||||||||
We are sad to report the passing of long-time contributor Alan Sare (Nov. 9, 1954 - Mar. 23, 2019). Alan's first contribution to Metamath was a shorter proof for tfrlem8 8398 in 2008. He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: https://us.metamath.org/other.html#completeusersproof 8398. His virtual deduction method is explained in the comment for wvd1 43931. Below are some excerpts from his first emails to NM in 2007: ...I have been interested in proving set theory theorems for many years for mental exercise. I enjoy it. I have used a book by Martin Zuckerman. It is informal. I am interested in completely and perfectly proving theorems. Mr. Zuckerman leaves out most of the steps of a proof, of course, like most authors do, as you have noted. A complete proof for higher theorems would require a volume of writing similar to the Metamath documents. So I am frustrated when I am not capable of constructing a proof and Zuckerman leaves out steps I do not understand. I could search for the steps in other texts, but I don't do that too much. Metamath may be the answer for me.... ...If we go beyond mathematics, I believe that it is possible to write down all human knowledge in a way similar to the way you have explicated large areas of mathematics. Of course, that would be a much, much more difficult job. For example, it is possible to take a hard science like physics, construct axioms based on experimental results, and to cast all of physics into a collection of axioms and theorems. Maybe this has already been attempted, although I am not familiar with it. When one then moves on to the soft sciences such as social science, this job gets much more difficult. The key is: All human thought consists of logical operations on abstract objects. Usually, these logical operations are done informally. There is no reason why one cannot take any subject and explicate it and take it down to the indivisible postulates in a formal rigorous way.... ...When I read a math book or an engineering book I come across something I don't understand and I am compelled to understand it. But, often it is hopeless. I don't have the time. Or, I would have to read the same thing by multiple authors in the hope that different authors would give parts of the working proof that others have omitted. It is very inefficient. Because I have always been inclined to "get to the bottom" for a 100% fully understood proof.... | ||||||||||||||||||||||||||||||||
Theorem | idiALT 43839 | Placeholder for idi 1. Though unnecessary, this theorem is sometimes used in proofs in this mathbox for pedagogical purposes. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ๐ โ โข ๐ | ||||||||||||||||||||||||||||||||
Theorem | exbir 43840 | Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 44215. (Contributed by Alan Sare, 31-Dec-2011.) | ||||||||||||||||||||||||||||||
โข (((๐ โง ๐) โ (๐ โ ๐)) โ (๐ โ (๐ โ (๐ โ ๐)))) | ||||||||||||||||||||||||||||||||
Theorem | 3impexpbicom 43841 | Version of 3impexp 1356 where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011.) | ||||||||||||||||||||||||||||||
โข (((๐ โง ๐ โง ๐) โ (๐ โ ๐)) โ (๐ โ (๐ โ (๐ โ (๐ โ ๐))))) | ||||||||||||||||||||||||||||||||
Theorem | 3impexpbicomi 43842 | Inference associated with 3impexpbicom 43841. Derived automatically from 3impexpbicomiVD 44220. (Contributed by Alan Sare, 31-Dec-2011.) | ||||||||||||||||||||||||||||||
โข ((๐ โง ๐ โง ๐) โ (๐ โ ๐)) โ โข (๐ โ (๐ โ (๐ โ (๐ โ ๐)))) | ||||||||||||||||||||||||||||||||
Theorem | bi1imp 43843 | Importation inference similar to imp 406, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ ๐)) โ โข ((๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi2imp 43844 | Importation inference similar to imp 406, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ ๐)) โ โข ((๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi3impb 43845 | Similar to 3impb 1113 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข ((๐ โง (๐ โง ๐)) โ ๐) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi3impa 43846 | Similar to 3impa 1108 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (((๐ โง ๐) โง ๐) โ ๐) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi23impib 43847 | 3impib 1114 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ ((๐ โง ๐) โ ๐)) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi13impib 43848 | 3impib 1114 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ ((๐ โง ๐) โ ๐)) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi123impib 43849 | 3impib 1114 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ ((๐ โง ๐) โ ๐)) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi13impia 43850 | 3impia 1115 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข ((๐ โง ๐) โ (๐ โ ๐)) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi123impia 43851 | 3impia 1115 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข ((๐ โง ๐) โ (๐ โ ๐)) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi33imp12 43852 | 3imp 1109 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi23imp13 43853 | 3imp 1109 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi13imp23 43854 | 3imp 1109 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi13imp2 43855 | Similar to 3imp 1109 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi12imp3 43856 | Similar to 3imp 1109 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi23imp1 43857 | Similar to 3imp 1109 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | bi123imp0 43858 | Similar to 3imp 1109 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) โ โข ((๐ โง ๐ โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | 4animp1 43859 | A single hypothesis unification deduction with an assertion which is an implication with a 4-right-nested conjunction antecedent. (Contributed by Alan Sare, 30-May-2018.) | ||||||||||||||||||||||||||||||
โข ((๐ โง ๐ โง ๐) โ (๐ โ ๐)) โ โข ((((๐ โง ๐) โง ๐) โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | 4an31 43860 | A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.) | ||||||||||||||||||||||||||||||
โข ((((๐ โง ๐) โง ๐) โง ๐) โ ๐) โ โข ((((๐ โง ๐) โง ๐) โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | 4an4132 43861 | A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.) | ||||||||||||||||||||||||||||||
โข ((((๐ โง ๐) โง ๐) โง ๐) โ ๐) โ โข ((((๐ โง ๐) โง ๐) โง ๐) โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | expcomdg 43862 | Biconditional form of expcomd 416. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ โ ((๐ โง ๐) โ ๐)) โ (๐ โ (๐ โ (๐ โ ๐)))) | ||||||||||||||||||||||||||||||||
Theorem | iidn3 43863 | idn3 43977 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) | ||||||||||||||||||||||||||||||||
Theorem | ee222 43864 | e222 43998 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ ๐)) & โข (๐ โ (๐ โ ๐)) & โข (๐ โ (๐ โ ๐)) & โข (๐ โ (๐ โ (๐ โ ๐))) โ โข (๐ โ (๐ โ ๐)) | ||||||||||||||||||||||||||||||||
Theorem | ee3bir 43865 | Right-biconditional form of e3 44099 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ ๐) โ โข (๐ โ (๐ โ (๐ โ ๐))) | ||||||||||||||||||||||||||||||||
Theorem | ee13 43866 | e13 44110 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ ๐) & โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ (๐ โ ๐)) โ โข (๐ โ (๐ โ (๐ โ ๐))) | ||||||||||||||||||||||||||||||||
Theorem | ee121 43867 | e121 44018 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ ๐) & โข (๐ โ (๐ โ ๐)) & โข (๐ โ ๐) & โข (๐ โ (๐ โ (๐ โ ๐))) โ โข (๐ โ (๐ โ ๐)) | ||||||||||||||||||||||||||||||||
Theorem | ee122 43868 | e122 44015 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ ๐) & โข (๐ โ (๐ โ ๐)) & โข (๐ โ (๐ โ ๐)) & โข (๐ โ (๐ โ (๐ โ ๐))) โ โข (๐ โ (๐ โ ๐)) | ||||||||||||||||||||||||||||||||
Theorem | ee333 43869 | e333 44095 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ (๐ โ (๐ โ ๐))) โ โข (๐ โ (๐ โ (๐ โ ๐))) | ||||||||||||||||||||||||||||||||
Theorem | ee323 43870 | e323 44128 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ (๐ โ ๐)) & โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ (๐ โ (๐ โ ๐))) โ โข (๐ โ (๐ โ (๐ โ ๐))) | ||||||||||||||||||||||||||||||||
Theorem | 3ornot23 43871 | If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 44209. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((ยฌ ๐ โง ยฌ ๐) โ ((๐ โจ ๐ โจ ๐) โ ๐)) | ||||||||||||||||||||||||||||||||
Theorem | orbi1r 43872 | orbi1 916 with order of disjuncts reversed. Derived from orbi1rVD 44210. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ โ ๐) โ ((๐ โจ ๐) โ (๐ โจ ๐))) | ||||||||||||||||||||||||||||||||
Theorem | 3orbi123 43873 | pm4.39 975 with a 3-conjunct antecedent. This proof is 3orbi123VD 44212 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (((๐ โ ๐) โง (๐ โ ๐) โง (๐ โ ๐)) โ ((๐ โจ ๐ โจ ๐) โ (๐ โจ ๐ โจ ๐))) | ||||||||||||||||||||||||||||||||
Theorem | syl5imp 43874 | Closed form of syl5 34. Derived automatically from syl5impVD 44225. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ โ (๐ โ ๐)) โ ((๐ โ ๐) โ (๐ โ (๐ โ ๐)))) | ||||||||||||||||||||||||||||||||
Theorem | impexpd 43875 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the
User's Proof was completed, it was minimized. The completed User's Proof
before minimization is not shown. (Contributed by Alan Sare,
18-Mar-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
| ||||||||||||||||||||||||||||||
โข ((๐ โ ((๐ โง ๐) โ ๐)) โ (๐ โ (๐ โ (๐ โ ๐)))) | ||||||||||||||||||||||||||||||||
Theorem | com3rgbi 43876 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual Deduction Proof (not shown) was minimized. The
minimized proof is shown.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
| ||||||||||||||||||||||||||||||
โข ((๐ โ (๐ โ (๐ โ ๐))) โ (๐ โ (๐ โ (๐ โ ๐)))) | ||||||||||||||||||||||||||||||||
Theorem | impexpdcom 43877 |
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual Deduction Proof (not shown) was minimized. The
minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
| ||||||||||||||||||||||||||||||
โข ((๐ โ ((๐ โง ๐) โ ๐)) โ (๐ โ (๐ โ (๐ โ ๐)))) | ||||||||||||||||||||||||||||||||
Theorem | ee1111 43878 |
Non-virtual deduction form of e1111 44037. (Contributed by Alan Sare,
18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program completeusersproof.cmd,
which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. The completed Virtual Deduction Proof (not shown) was
minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||
โข (๐ โ ๐) & โข (๐ โ ๐) & โข (๐ โ ๐) & โข (๐ โ ๐) & โข (๐ โ (๐ โ (๐ โ (๐ โ ๐)))) โ โข (๐ โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | pm2.43bgbi 43879 |
Logical equivalence of a 2-left-nested implication and a 1-left-nested
implicated
when two antecedents of the former implication are identical.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual
Deduction Proof (not shown) was minimized. The minimized proof is
shown.
| ||||||||||||||||||||||||||||||
โข ((๐ โ (๐ โ (๐ โ ๐))) โ (๐ โ (๐ โ ๐))) | ||||||||||||||||||||||||||||||||
Theorem | pm2.43cbi 43880 |
Logical equivalence of a 3-left-nested implication and a 2-left-nested
implicated when two antecedents of the former implication are identical.
(Contributed by Alan Sare, 18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is
a Virtual Deduction proof completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof
(not shown) was minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||
โข ((๐ โ (๐ โ (๐ โ (๐ โ ๐)))) โ (๐ โ (๐ โ (๐ โ ๐)))) | ||||||||||||||||||||||||||||||||
Theorem | ee233 43881 |
Non-virtual deduction form of e233 44127. (Contributed by Alan Sare,
18-Mar-2012.)
(Proof modification is discouraged.) (New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof completed
automatically by the tools program completeusersproof.cmd, which invokes
Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The
completed Virtual
Deduction Proof (not shown) was minimized. The minimized proof is
shown.
| ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ ๐)) & โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ (๐ โ (๐ โ ๐))) โ โข (๐ โ (๐ โ (๐ โ ๐))) | ||||||||||||||||||||||||||||||||
Theorem | imbi13 43882 | Join three logical equivalences to form equivalence of implications. imbi13 43882 is imbi13VD 44236 without virtual deductions and was automatically derived from imbi13VD 44236 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ โ ๐) โ ((๐ โ ๐) โ ((๐ โ ๐) โ ((๐ โ (๐ โ ๐)) โ (๐ โ (๐ โ ๐)))))) | ||||||||||||||||||||||||||||||||
Theorem | ee33 43883 |
Non-virtual deduction form of e33 44096. (Contributed by Alan Sare,
18-Mar-2012.) (Proof modification is discouraged.)
(New usage is discouraged.)
The following User's Proof is a Virtual Deduction proof
completed automatically by the tools program completeusersproof.cmd,
which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. The completed Virtual Deduction Proof (not shown) was
minimized. The minimized proof is shown.
| ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ (๐ โ (๐ โ ๐))) & โข (๐ โ (๐ โ ๐)) โ โข (๐ โ (๐ โ (๐ โ ๐))) | ||||||||||||||||||||||||||||||||
Theorem | con5 43884 | Biconditional contraposition variation. This proof is con5VD 44262 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ โ ยฌ ๐) โ (ยฌ ๐ โ ๐)) | ||||||||||||||||||||||||||||||||
Theorem | con5i 43885 | Inference form of con5 43884. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ ยฌ ๐) โ โข (ยฌ ๐ โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | exlimexi 43886 | Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ โ๐ฅ๐) & โข (โ๐ฅ๐ โ (๐ โ ๐)) โ โข (โ๐ฅ๐ โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | sb5ALT 43887* | Equivalence for substitution. Alternate proof of sb5 2260. This proof is sb5ALTVD 44275 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ([๐ฆ / ๐ฅ]๐ โ โ๐ฅ(๐ฅ = ๐ฆ โง ๐)) | ||||||||||||||||||||||||||||||||
Theorem | eexinst01 43888 | exinst01 43987 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข โ๐ฅ๐ & โข (๐ โ (๐ โ ๐)) & โข (๐ โ โ๐ฅ๐) & โข (๐ โ โ๐ฅ๐) โ โข (๐ โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | eexinst11 43889 | exinst11 43988 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ โ๐ฅ๐) & โข (๐ โ (๐ โ ๐)) & โข (๐ โ โ๐ฅ๐) & โข (๐ โ โ๐ฅ๐) โ โข (๐ โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | vk15.4j 43890 | Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 43890 is vk15.4jVD 44276 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ยฌ (โ๐ฅ ยฌ ๐ โง โ๐ฅ(๐ โง ยฌ ๐)) & โข (โ๐ฅ๐ โ ยฌ โ๐ฅ(๐ โง ๐)) & โข ยฌ โ๐ฅ(๐ โ ๐) โ โข (ยฌ โ๐ฅ ยฌ ๐ โ ยฌ โ๐ฅ๐) | ||||||||||||||||||||||||||||||||
Theorem | notnotrALT 43891 | Converse of double negation. Alternate proof of notnotr 130. This proof is notnotrALTVD 44277 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (ยฌ ยฌ ๐ โ ๐) | ||||||||||||||||||||||||||||||||
Theorem | con3ALT2 43892 | Contraposition. Alternate proof of con3 153. This proof is con3ALTVD 44278 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ โ ๐) โ (ยฌ ๐ โ ยฌ ๐)) | ||||||||||||||||||||||||||||||||
Theorem | ssralv2 43893* | Quantification restricted to a subclass for two quantifiers. ssralv 4046 for two quantifiers. The proof of ssralv2 43893 was automatically generated by minimizing the automatically translated proof of ssralv2VD 44228. The automatic translation is by the tools program translate_without_overwriting.cmd. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ด โ ๐ต โง ๐ถ โ ๐ท) โ (โ๐ฅ โ ๐ต โ๐ฆ โ ๐ท ๐ โ โ๐ฅ โ ๐ด โ๐ฆ โ ๐ถ ๐)) | ||||||||||||||||||||||||||||||||
Theorem | sbc3or 43894 | sbcor 3827 with a 3-disjuncts. This proof is sbc3orgVD 44213 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ([๐ด / ๐ฅ](๐ โจ ๐ โจ ๐) โ ([๐ด / ๐ฅ]๐ โจ [๐ด / ๐ฅ]๐ โจ [๐ด / ๐ฅ]๐)) | ||||||||||||||||||||||||||||||||
Theorem | alrim3con13v 43895* | Closed form of alrimi 2199 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 44214 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ โ โ๐ฅ๐) โ ((๐ โง ๐ โง ๐) โ โ๐ฅ(๐ โง ๐ โง ๐))) | ||||||||||||||||||||||||||||||||
Theorem | rspsbc2 43896* | rspsbc 3869 with two quantifying variables. This proof is rspsbc2VD 44217 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ด โ ๐ต โ (๐ถ โ ๐ท โ (โ๐ฅ โ ๐ต โ๐ฆ โ ๐ท ๐ โ [๐ถ / ๐ฆ][๐ด / ๐ฅ]๐))) | ||||||||||||||||||||||||||||||||
Theorem | sbcoreleleq 43897* | Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 44221. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ด โ ๐ โ ([๐ด / ๐ฆ](๐ฅ โ ๐ฆ โจ ๐ฆ โ ๐ฅ โจ ๐ฅ = ๐ฆ) โ (๐ฅ โ ๐ด โจ ๐ด โ ๐ฅ โจ ๐ฅ = ๐ด))) | ||||||||||||||||||||||||||||||||
Theorem | tratrb 43898* | If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 44223. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((Tr ๐ด โง โ๐ฅ โ ๐ด โ๐ฆ โ ๐ด (๐ฅ โ ๐ฆ โจ ๐ฆ โ ๐ฅ โจ ๐ฅ = ๐ฆ) โง ๐ต โ ๐ด) โ Tr ๐ต) | ||||||||||||||||||||||||||||||||
Theorem | ordelordALT 43899 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6385 using the Axiom of Regularity indirectly through dford2 9635. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that E Fr ๐ด because this is inferred by the Axiom of Regularity. ordelordALT 43899 is ordelordALTVD 44229 without virtual deductions and was automatically derived from ordelordALTVD 44229 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((Ord ๐ด โง ๐ต โ ๐ด) โ Ord ๐ต) | ||||||||||||||||||||||||||||||||
Theorem | sbcim2g 43900 | Distribution of class substitution over a left-nested implication. Similar to sbcimg 3825. sbcim2g 43900 is sbcim2gVD 44237 without virtual deductions and was automatically derived from sbcim2gVD 44237 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ด โ ๐ โ ([๐ด / ๐ฅ](๐ โ (๐ โ ๐)) โ ([๐ด / ๐ฅ]๐ โ ([๐ด / ๐ฅ]๐ โ [๐ด / ๐ฅ]๐)))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |