![]() |
Metamath
Proof Explorer Theorem List (p. 433 of 479) | < 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: | ![]() (1-30166) |
![]() (30167-31689) |
![]() (31690-47842) |
Type | Label | Description | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Statement | ||||||||||||||||||||||||||||||||
Theorem | fvsb 43201* | 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 43202* | 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 43203 | 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 43204 | 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 9586 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 43205 | 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 43206 | Introduce the operation of vector addition. | ||||||||||||||||||||||||||||||
class +๐ | ||||||||||||||||||||||||||||||||
Syntax | cminusr 43207 | Introduce the operation of vector subtraction. | ||||||||||||||||||||||||||||||
class -๐ | ||||||||||||||||||||||||||||||||
Syntax | ctimesr 43208 | Introduce the operation of scalar multiplication. | ||||||||||||||||||||||||||||||
class .๐ฃ | ||||||||||||||||||||||||||||||||
Syntax | cptdfc 43209 | 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 43210 | RR3 is a class. | ||||||||||||||||||||||||||||||
class RR3 | ||||||||||||||||||||||||||||||||
Syntax | cline3 43211 | line3 is a class. | ||||||||||||||||||||||||||||||
class line3 | ||||||||||||||||||||||||||||||||
Definition | df-addr 43212* | 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 43213* | 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 43214* | 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 43215* | 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 43216* | 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 43217* | 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 43218 | 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 43219 | 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 43220 | 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 43221 | 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 43222 | 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 43223 | 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 43224 | 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 43225* | 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 43226 | 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 43227* | 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 8383 in 2008. He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: https://us.metamath.org/other.html#completeusersproof 8383. His virtual deduction method is explained in the comment for wvd1 43320. 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 43228 | 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 43229 | Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 43604. (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 43230 | 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 43231 | Inference associated with 3impexpbicom 43230. Derived automatically from 3impexpbicomiVD 43609. (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 43232 | 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 43233 | 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 43234 | 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 43235 | 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 43236 | 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 43237 | 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 43238 | 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 43239 | 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 43240 | 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 43241 | 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 43242 | 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 43243 | 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 43244 | 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 43245 | 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 43246 | 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 43247 | 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 43248 | 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 43249 | 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 43250 | 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 43251 | 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 43252 | idn3 43366 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 43253 | e222 43387 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 43254 | Right-biconditional form of e3 43488 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 43255 | e13 43499 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 43256 | e121 43407 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 43257 | e122 43404 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 43258 | e333 43484 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 43259 | e323 43517 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 43260 | If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 43598. (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 43261 | orbi1 916 with order of disjuncts reversed. Derived from orbi1rVD 43599. (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 43262 | pm4.39 975 with a 3-conjunct antecedent. This proof is 3orbi123VD 43601 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 43263 | Closed form of syl5 34. Derived automatically from syl5impVD 43614. (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 43264 |
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 43265 |
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 43266 |
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 43267 |
Non-virtual deduction form of e1111 43426. (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 43268 |
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 43269 |
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 43270 |
Non-virtual deduction form of e233 43516. (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 43271 | Join three logical equivalences to form equivalence of implications. imbi13 43271 is imbi13VD 43625 without virtual deductions and was automatically derived from imbi13VD 43625 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 43272 |
Non-virtual deduction form of e33 43485. (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 43273 | Biconditional contraposition variation. This proof is con5VD 43651 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 43274 | Inference form of con5 43273. (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 43275 | 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 43276* | Equivalence for substitution. Alternate proof of sb5 2267. This proof is sb5ALTVD 43664 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 43277 | exinst01 43376 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 43278 | exinst11 43377 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 43279 | 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 43279 is vk15.4jVD 43665 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 43280 | Converse of double negation. Alternate proof of notnotr 130. This proof is notnotrALTVD 43666 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 43281 | Contraposition. Alternate proof of con3 153. This proof is con3ALTVD 43667 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 43282* | Quantification restricted to a subclass for two quantifiers. ssralv 4050 for two quantifiers. The proof of ssralv2 43282 was automatically generated by minimizing the automatically translated proof of ssralv2VD 43617. 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 43283 | sbcor 3830 with a 3-disjuncts. This proof is sbc3orgVD 43602 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 43284* | Closed form of alrimi 2206 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 43603 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 43285* | rspsbc 3873 with two quantifying variables. This proof is rspsbc2VD 43606 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 43286* | Substitution of a setvar variable for another setvar variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 43610. (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 43287* | 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 43612. (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 43288 | 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 9614. 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 43288 is ordelordALTVD 43618 without virtual deductions and was automatically derived from ordelordALTVD 43618 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 43289 | Distribution of class substitution over a left-nested implication. Similar to sbcimg 3828. sbcim2g 43289 is sbcim2gVD 43626 without virtual deductions and was automatically derived from sbcim2gVD 43626 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 43290 | Implication form of sbcbii 3837. sbcbi 43290 is sbcbiVD 43627 without virtual deductions and was automatically derived from sbcbiVD 43627 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 43291* | Formula-building inference rule for class substitution, substituting a class variable for the setvar variable of the transitivity predicate. trsbc 43291 is trsbcVD 43628 without virtual deductions and was automatically derived from trsbcVD 43628 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 43292* | The union of a class of transitive sets is transitive. Alternate proof of truni 5281. truniALT 43292 is truniALTVD 43629 without virtual deductions and was automatically derived from truniALTVD 43629 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 43293* | Lemma for onfrALT 43300. (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 43294* | Lemma for onfrALT 43300. (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 43295* | Lemma for onfrALT 43300. (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 43296* | gen31 43372 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 43297* | Lemma for onfrALT 43300. (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 43298* | 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 43299* | Lemma for onfrALT 43300. (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 43300 | The membership relation is foundational on the class of ordinal numbers. onfrALT 43300 is an alternate proof of onfr 6403. onfrALTVD 43642 is the Virtual Deduction proof from which onfrALT 43300 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 43642. 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 |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |