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Type | Label | Description | ||||||||||||||||||||||||||||||
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Statement | ||||||||||||||||||||||||||||||||
Theorem | xpexb 43301 | 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 43302 | An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5660, ax-reg 9589 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 43303 | 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 43304 | Introduce the operation of vector addition. | ||||||||||||||||||||||||||||||
class +๐ | ||||||||||||||||||||||||||||||||
Syntax | cminusr 43305 | Introduce the operation of vector subtraction. | ||||||||||||||||||||||||||||||
class -๐ | ||||||||||||||||||||||||||||||||
Syntax | ctimesr 43306 | Introduce the operation of scalar multiplication. | ||||||||||||||||||||||||||||||
class .๐ฃ | ||||||||||||||||||||||||||||||||
Syntax | cptdfc 43307 | 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 43308 | RR3 is a class. | ||||||||||||||||||||||||||||||
class RR3 | ||||||||||||||||||||||||||||||||
Syntax | cline3 43309 | line3 is a class. | ||||||||||||||||||||||||||||||
class line3 | ||||||||||||||||||||||||||||||||
Definition | df-addr 43310* | 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 43311* | 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 43312* | 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 43313* | 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 43314* | 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 43315* | 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 43316 | 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 43317 | 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 43318 | 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 43319 | 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 43320 | 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 43321 | 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 43322 | 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 43323* | 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 43324 | 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 43325* | 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 8386 in 2008. He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: https://us.metamath.org/other.html#completeusersproof 8386. His virtual deduction method is explained in the comment for wvd1 43418. 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 43326 | 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 43327 | Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 43702. (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 43328 | Version of 3impexp 1358 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 43329 | Inference associated with 3impexpbicom 43328. Derived automatically from 3impexpbicomiVD 43707. (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 43330 | Importation inference similar to imp 407, 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 43331 | Importation inference similar to imp 407, 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 43332 | Similar to 3impb 1115 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 43333 | Similar to 3impa 1110 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 43334 | 3impib 1116 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 43335 | 3impib 1116 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 43336 | 3impib 1116 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 43337 | 3impia 1117 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 43338 | 3impia 1117 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 43339 | 3imp 1111 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 43340 | 3imp 1111 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 43341 | 3imp 1111 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 43342 | Similar to 3imp 1111 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 43343 | Similar to 3imp 1111 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 43344 | Similar to 3imp 1111 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 43345 | Similar to 3imp 1111 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 43346 | 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 43347 | 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 43348 | 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 43349 | Biconditional form of expcomd 417. (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 43350 | idn3 43464 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 43351 | e222 43485 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 43352 | Right-biconditional form of e3 43586 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 43353 | e13 43597 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 43354 | e121 43505 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 43355 | e122 43502 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 43356 | e333 43582 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 43357 | e323 43615 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 43358 | If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 43696. (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 43359 | orbi1 916 with order of disjuncts reversed. Derived from orbi1rVD 43697. (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 43360 | pm4.39 975 with a 3-conjunct antecedent. This proof is 3orbi123VD 43699 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 43361 | Closed form of syl5 34. Derived automatically from syl5impVD 43712. (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 43362 |
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 43363 |
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 43364 |
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 43365 |
Non-virtual deduction form of e1111 43524. (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 43366 |
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 43367 |
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 43368 |
Non-virtual deduction form of e233 43614. (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 43369 | Join three logical equivalences to form equivalence of implications. imbi13 43369 is imbi13VD 43723 without virtual deductions and was automatically derived from imbi13VD 43723 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 43370 |
Non-virtual deduction form of e33 43583. (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 43371 | Biconditional contraposition variation. This proof is con5VD 43749 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 43372 | Inference form of con5 43371. (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 43373 | 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 43374* | Equivalence for substitution. Alternate proof of sb5 2267. This proof is sb5ALTVD 43762 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 43375 | exinst01 43474 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 43376 | exinst11 43475 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 43377 | 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 43377 is vk15.4jVD 43763 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 43378 | Converse of double negation. Alternate proof of notnotr 130. This proof is notnotrALTVD 43764 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 43379 | Contraposition. Alternate proof of con3 153. This proof is con3ALTVD 43765 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 43380* | Quantification restricted to a subclass for two quantifiers. ssralv 4050 for two quantifiers. The proof of ssralv2 43380 was automatically generated by minimizing the automatically translated proof of ssralv2VD 43715. 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 43381 | sbcor 3830 with a 3-disjuncts. This proof is sbc3orgVD 43700 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 43382* | Closed form of alrimi 2206 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 43701 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 43383* | rspsbc 3873 with two quantifying variables. This proof is rspsbc2VD 43704 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 43384* | Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 43708. (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 43385* | 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 43710. (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 43386 | An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6386 using the Axiom of Regularity indirectly through dford2 9617. 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 43386 is ordelordALTVD 43716 without virtual deductions and was automatically derived from ordelordALTVD 43716 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 43387 | Distribution of class substitution over a left-nested implication. Similar to sbcimg 3828. sbcim2g 43387 is sbcim2gVD 43724 without virtual deductions and was automatically derived from sbcim2gVD 43724 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)))) | ||||||||||||||||||||||||||||||||
Theorem | sbcbi 43388 | Implication form of sbcbii 3837. sbcbi 43388 is sbcbiVD 43725 without virtual deductions and was automatically derived from sbcbiVD 43725 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 (โ๐ฅ(๐ โ ๐) โ ([๐ด / ๐ฅ]๐ โ [๐ด / ๐ฅ]๐))) | ||||||||||||||||||||||||||||||||
Theorem | trsbc 43389* | Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 43389 is trsbcVD 43726 without virtual deductions and was automatically derived from trsbcVD 43726 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]Tr ๐ฅ โ Tr ๐ด)) | ||||||||||||||||||||||||||||||||
Theorem | truniALT 43390* | The union of a class of transitive sets is transitive. Alternate proof of truni 5281. truniALT 43390 is truniALTVD 43727 without virtual deductions and was automatically derived from truniALTVD 43727 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 (โ๐ฅ โ ๐ด Tr ๐ฅ โ Tr โช ๐ด) | ||||||||||||||||||||||||||||||||
Theorem | onfrALTlem5 43391* | Lemma for onfrALT 43398. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ([(๐ โฉ ๐ฅ) / ๐]((๐ โ (๐ โฉ ๐ฅ) โง ๐ โ โ ) โ โ๐ฆ โ ๐ (๐ โฉ ๐ฆ) = โ ) โ (((๐ โฉ ๐ฅ) โ (๐ โฉ ๐ฅ) โง (๐ โฉ ๐ฅ) โ โ ) โ โ๐ฆ โ (๐ โฉ ๐ฅ)((๐ โฉ ๐ฅ) โฉ ๐ฆ) = โ )) | ||||||||||||||||||||||||||||||||
Theorem | onfrALTlem4 43392* | Lemma for onfrALT 43398. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ([๐ฆ / ๐ฅ](๐ฅ โ ๐ โง (๐ โฉ ๐ฅ) = โ ) โ (๐ฆ โ ๐ โง (๐ โฉ ๐ฆ) = โ )) | ||||||||||||||||||||||||||||||||
Theorem | onfrALTlem3 43393* | Lemma for onfrALT 43398. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ โ On โง ๐ โ โ ) โ ((๐ฅ โ ๐ โง ยฌ (๐ โฉ ๐ฅ) = โ ) โ โ๐ฆ โ (๐ โฉ ๐ฅ)((๐ โฉ ๐ฅ) โฉ ๐ฆ) = โ )) | ||||||||||||||||||||||||||||||||
Theorem | ggen31 43394* | gen31 43470 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (๐ โ (๐ โ (๐ โ ๐))) โ โข (๐ โ (๐ โ (๐ โ โ๐ฅ๐))) | ||||||||||||||||||||||||||||||||
Theorem | onfrALTlem2 43395* | Lemma for onfrALT 43398. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ โ On โง ๐ โ โ ) โ ((๐ฅ โ ๐ โง ยฌ (๐ โฉ ๐ฅ) = โ ) โ โ๐ฆ โ ๐ (๐ โฉ ๐ฆ) = โ )) | ||||||||||||||||||||||||||||||||
Theorem | cbvexsv 43396* | A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (โ๐ฅ๐ โ โ๐ฆ[๐ฆ / ๐ฅ]๐) | ||||||||||||||||||||||||||||||||
Theorem | onfrALTlem1 43397* | Lemma for onfrALT 43398. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข ((๐ โ On โง ๐ โ โ ) โ ((๐ฅ โ ๐ โง (๐ โฉ ๐ฅ) = โ ) โ โ๐ฆ โ ๐ (๐ โฉ ๐ฆ) = โ )) | ||||||||||||||||||||||||||||||||
Theorem | onfrALT 43398 | The membership relation is foundational on the class of ordinal numbers. onfrALT 43398 is an alternate proof of onfr 6403. onfrALTVD 43740 is the Virtual Deduction proof from which onfrALT 43398 is derived. The Virtual Deduction proof mirrors the working proof of onfr 6403 which is the main part of the proof of Theorem 7.12 of the first edition of TakeutiZaring. The proof of the corresponding Proposition 7.12 of [TakeutiZaring] p. 38 (second edition) does not contain the working proof equivalent of onfrALTVD 43740. This theorem does not rely on the Axiom of Regularity. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข E Fr On | ||||||||||||||||||||||||||||||||
Theorem | 19.41rg 43399 | Closed form of right-to-left implication of 19.41 2228, Theorem 19.41 of [Margaris] p. 90. Derived from 19.41rgVD 43751. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (โ๐ฅ(๐ โ โ๐ฅ๐) โ ((โ๐ฅ๐ โง ๐) โ โ๐ฅ(๐ โง ๐))) | ||||||||||||||||||||||||||||||||
Theorem | opelopab4 43400* | Ordered pair membership in a class abstraction of ordered pairs. Compare to elopab 5527. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||||||||||||||||||||
โข (โจ๐ข, ๐ฃโฉ โ {โจ๐ฅ, ๐ฆโฉ โฃ ๐} โ โ๐ฅโ๐ฆ((๐ฅ = ๐ข โง ๐ฆ = ๐ฃ) โง ๐)) |
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