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Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbcth2 3201* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremra5 3202 Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1812. (Contributed by NM, 16-Jan-2004.)

Theoremrmo2 3203* Alternate definition of restricted "at most one." Note that is not equivalent to (in analogy to reu6 3080); to see this, let be the empty set. However, one direction of this pattern holds; see rmo2i 3204. (Contributed by NM, 17-Jun-2017.)

Theoremrmo2i 3204* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)

Theoremrmo3 3205* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

Theoremrmob 3206* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)

Theoremrmoi 3207* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

2.1.10  Proper substitution of classes for sets into classes

Syntaxcsb 3208 Extend class notation to include the proper substitution of a class for a set into another class.

Definitiondf-csb 3209* Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 3118, to prevent ambiguity. Theorem sbcel1g 3227 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3236 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)

Theoremcsb2 3210* Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that can be free in but cannot occur in . (Contributed by NM, 2-Dec-2013.)

Theoremcsbeq1 3211 Analog of dfsbcq 3120 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcbvcsb 3212 Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on . (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremcbvcsbv 3213* Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbeq1d 3214 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)

Theoremcsbid 3215 Analog of sbid 1943 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbeq1a 3216 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbco 3217* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbexg 3218 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbex 3219 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremcsbtt 3220 Substitution doesn't affect a constant (in which is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremcsbconstgf 3221 Substitution doesn't affect a constant (in which is not free). (Contributed by NM, 10-Nov-2005.)

Theoremcsbconstg 3222* Substitution doesn't affect a constant (in which is not free). csbconstgf 3221 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)

Theoremsbcel12g 3223 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbceqg 3224 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcnel12g 3225 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)

Theoremsbcne12g 3226 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)

Theoremsbcel1g 3227* Move proper substitution in and out of a membership relation. Note that the scope of is the wff , whereas the scope of is the class . (Contributed by NM, 10-Nov-2005.)

Theoremsbceq1g 3228* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)

Theoremsbcel2g 3229* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)

Theoremsbceq2g 3230* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)

Theoremcsbcomg 3231* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)

Theoremcsbeq2d 3232 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2dv 3233* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2i 3234 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbvarg 3235 The proper substitution of a class for set variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)

Theoremsbccsbg 3236* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)

Theoremsbccsb2g 3237 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)

Theoremnfcsb1d 3238 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1 3239 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1v 3240* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsbd 3241 Deduction version of nfcsb 3242. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb 3242 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremcsbhypf 3243* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2958 for class substitution version. (Contributed by NM, 19-Dec-2008.)

Theoremcsbiebt 3244* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3248.) (Contributed by NM, 11-Nov-2005.)

Theoremcsbiedf 3245* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

Theoremcsbieb 3246* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)

Theoremcsbiebg 3247* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremcsbiegf 3248* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbief 3249* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbied 3250* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbied2 3251* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcsbie2t 3252* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3253). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbie2 3253* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)

Theoremcsbie2g 3254* Conversion of implicit substitution to explicit class substitution. This version of sbcie 3152 avoids a disjointness condition on by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremsbcnestgf 3255 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestgf 3256 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Theoremsbcnestg 3257* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestg 3258* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

TheoremcsbnestgOLD 3259* Nest the composition of two substitutions. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 23-Nov-2005.)

Theoremcsbnest1g 3260 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremcsbidmg 3261* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)

Theoremsbcco3g 3262* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Theoremsbcco3gOLD 3263* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremcsbco3g 3264* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Theoremrspcsbela 3265* Special case related to rspsbc 3196. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)

Theoremsbnfc2 3266* Two ways of expressing " is (effectively) not free in ." (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremcsbabg 3267* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremcbvralcsf 3268 A more general version of cbvralf 2883 that doesn't require and to be distinct from or . Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvrexcsf 3269 A more general version of cbvrexf 2884 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)

Theoremcbvreucsf 3270 A more general version of cbvreuv 2891 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvrabcsf 3271 A more general version of cbvrab 2911 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvralv2 3272* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

Theoremcbvrexv2 3273* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

2.1.11  Define basic set operations and relations

Syntaxcdif 3274 Extend class notation to include class difference (read: " minus ").

Syntaxcun 3275 Extend class notation to include union of two classes (read: " union ").

Syntaxcin 3276 Extend class notation to include the intersection of two classes (read: " intersect ").

Syntaxwss 3277 Extend wff notation to include the subclass relation. This is read " is a subclass of " or " includes ." When exists as a set, it is also read " is a subset of ."

Syntaxwpss 3278 Extend wff notation with proper subclass relation.

Theoremdifjust 3279* Soundness justification theorem for df-dif 3280. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-dif 3280* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, (ex-dif 21593). Contrast this operation with union (df-un 3282) and intersection (df-in 3284). Several notations are used in the literature; we chose the convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology " excludes " to mean . We will use " is removed from " to mean i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)

Theoremunjust 3281* Soundness justification theorem for df-un 3282. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-un 3282* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, (ex-un 21594). Contrast this operation with difference (df-dif 3280) and intersection (df-in 3284). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 3533. For union defined in terms of intersection, see dfun3 3536. (Contributed by NM, 23-Aug-1993.)

Theoreminjust 3283* Soundness justification theorem for df-in 3284. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-in 3284* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, (ex-in 21595). Contrast this operation with union (df-un 3282) and difference (df-dif 3280). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3534 and dfin4 3538. For intersection defined in terms of union, see dfin3 3537. (Contributed by NM, 29-Apr-1994.)

Theoremdfin5 3285* Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)

Theoremdfdif2 3286* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)

Theoremeldif 3287 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)

Theoremeldifd 3288 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3287. (Contributed by David Moews, 1-May-2017.)

Theoremeldifad 3289 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3287. (Contributed by David Moews, 1-May-2017.)

Theoremeldifbd 3290 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3287. (Contributed by David Moews, 1-May-2017.)

2.1.12  Subclasses and subsets

Definitiondf-ss 3291 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example, (ex-ss 21597). Note that (proved in ssid 3324). Contrast this relationship with the relationship (as will be defined in df-pss 3293). For a more traditional definition, but requiring a dummy variable, see dfss2 3294. Other possible definitions are given by dfss3 3295, dfss4 3532, sspss 3403, ssequn1 3474, ssequn2 3477, sseqin2 3517, and ssdif0 3643. (Contributed by NM, 27-Apr-1994.)

Theoremdfss 3292 Variant of subclass definition df-ss 3291. (Contributed by NM, 3-Sep-2004.)

Definitiondf-pss 3293 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. For example, (ex-pss 21598). Note that (proved in pssirr 3404). Contrast this relationship with the relationship (as defined in df-ss 3291). Other possible definitions are given by dfpss2 3389 and dfpss3 3390. (Contributed by NM, 7-Feb-1996.)

Theoremdfss2 3294* Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)

Theoremdfss3 3295* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)

Theoremdfss2f 3296 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)

Theoremdfss3f 3297 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)

Theoremnfss 3298 If is not free in and , it is not free in . (Contributed by NM, 27-Dec-1996.)

Theoremssel 3299 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)

Theoremssel2 3300 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)

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