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

Theoremssdifss 3201 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)

Theoremddifnel 3202* Double complement under universal class. The hypothesis corresponds to stability of membership in , which is weaker than decidability (see dcstab 829). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3203) . Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that is a subset of , see ddifss 3309. (Contributed by Jim Kingdon, 21-Jul-2018.)

Theoremddifstab 3203* A class is equal to its double complement if and only if it is stable (that is, membership in it is a stable property). (Contributed by BJ, 12-Dec-2021.)
STAB

Theoremssconb 3204 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)

Theoremsscon 3205 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)

Theoremssdif 3206 Difference law for subsets. (Contributed by NM, 28-May-1998.)

Theoremssdifd 3207 If is contained in , then is contained in . Deduction form of ssdif 3206. (Contributed by David Moews, 1-May-2017.)

Theoremsscond 3208 If is contained in , then is contained in . Deduction form of sscon 3205. (Contributed by David Moews, 1-May-2017.)

Theoremssdifssd 3209 If is contained in , then is also contained in . Deduction form of ssdifss 3201. (Contributed by David Moews, 1-May-2017.)

Theoremssdif2d 3210 If is contained in and is contained in , then is contained in . Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremraldifb 3211 Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)

2.1.13.2  The union of two classes

Theoremelun 3212 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)

Theoremuneqri 3213* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)

Theoremunidm 3214 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremuncom 3215 Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremequncom 3216 If a class equals the union of two other classes, then it equals the union of those two classes commuted. (Contributed by Alan Sare, 18-Feb-2012.)

Theoremequncomi 3217 Inference form of equncom 3216. (Contributed by Alan Sare, 18-Feb-2012.)

Theoremuneq1 3218 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)

Theoremuneq2 3219 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)

Theoremuneq12 3220 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)

Theoremuneq1i 3221 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)

Theoremuneq2i 3222 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)

Theoremuneq12i 3223 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremuneq1d 3224 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)

Theoremuneq2d 3225 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)

Theoremuneq12d 3226 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnfun 3227 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremunass 3228 Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremun12 3229 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)

Theoremun23 3230 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremun4 3231 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)

Theoremunundi 3232 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremunundir 3233 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremssun1 3234 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremssun2 3235 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)

Theoremssun3 3236 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)

Theoremssun4 3237 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)

Theoremelun1 3238 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)

Theoremelun2 3239 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)

Theoremunss1 3240 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssequn1 3241 A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremunss2 3242 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)

Theoremunss12 3243 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)

Theoremssequn2 3244 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)

Theoremunss 3245 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)

Theoremunssi 3246 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremunssd 3247 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremunssad 3248 If is contained in , so is . One-way deduction form of unss 3245. Partial converse of unssd 3247. (Contributed by David Moews, 1-May-2017.)

Theoremunssbd 3249 If is contained in , so is . One-way deduction form of unss 3245. Partial converse of unssd 3247. (Contributed by David Moews, 1-May-2017.)

Theoremssun 3250 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)

Theoremrexun 3251 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremralunb 3252 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremralun 3253 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)

2.1.13.3  The intersection of two classes

Theoremelin 3254 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)

Theoremelini 3255 Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Theoremelind 3256 Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremelinel1 3257 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremelinel2 3258 Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremelin2 3259 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Theoremelin1d 3260 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)

Theoremelin2d 3261 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)

Theoremelin3 3262 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Theoremincom 3263 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)

Theoremineqri 3264* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)

Theoremineq1 3265 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)

Theoremineq2 3266 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq12 3267 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)

Theoremineq1i 3268 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq2i 3269 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq12i 3270 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremineq1d 3271 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Theoremineq2d 3272 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Theoremineq12d 3273 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremineqan12d 3274 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)

Theoremdfss1 3275 A frequently-used variant of subclass definition df-ss 3079. (Contributed by NM, 10-Jan-2015.)

Theoremdfss5 3276 Another definition of subclasshood. Similar to df-ss 3079, dfss 3080, and dfss1 3275. (Contributed by David Moews, 1-May-2017.)

Theoremnfin 3277 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcsbing 3278 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)

Theoremrabbi2dva 3279* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)

Theoreminidm 3280 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Theoreminass 3281 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)

Theoremin12 3282 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)

Theoremin32 3283 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremin13 3284 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Theoremin31 3285 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Theoreminrot 3286 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)

Theoremin4 3287 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)

Theoreminindi 3288 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)

Theoreminindir 3289 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremsseqin2 3290 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)

Theoreminss1 3291 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)

Theoreminss2 3292 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)

Theoremssin 3293 Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssini 3294 An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)

Theoremssind 3295 A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremssrin 3296 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremsslin 3297 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)

Theoremssrind 3298 Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Theoremss2in 3299 Intersection of subclasses. (Contributed by NM, 5-May-2000.)

Theoremssinss1 3300 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)

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