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

Theoremsseq12 3201 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)

Theoremsseq1i 3202 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)

Theoremsseq2i 3203 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)

Theoremsseq12i 3204 An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremsseq1d 3205 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

Theoremsseq2d 3206 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

Theoremsseq12d 3207 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)

Theoremeqsstri 3208 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)

Theoremeqsstr3i 3209 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)

Theoremsseqtri 3210 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)

Theoremsseqtr4i 3211 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)

Theoremeqsstrd 3212 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremeqsstr3d 3213 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtrd 3214 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtr4d 3215 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theorem3sstr3i 3216 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theorem3sstr4i 3217 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theorem3sstr3g 3218 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4g 3219 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theorem3sstr3d 3220 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4d 3221 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremsyl5eqss 3222 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl5eqssr 3223 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl6sseq 3224 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl6sseqr 3225 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl5sseq 3226 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsyl5sseqr 3227 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsyl6eqss 3228 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremsyl6eqssr 3229 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremeqimss 3230 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremeqimss2 3231 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)

Theoremeqimssi 3232 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)

Theoremeqimss2i 3233 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)

Theoremnssne1 3234 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnssne2 3235 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnss 3236* Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremssralv 3237* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)

Theoremssrexv 3238* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)

Theoremralss 3239* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremrexss 3240* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremss2ab 3241 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)

Theoremabss 3242* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssab 3243* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssabral 3244* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)

Theoremss2abi 3245 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)

Theoremss2abdv 3246* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)

Theoremabssdv 3247* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremabssi 3248* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremss2rab 3249 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)

Theoremrabss 3250* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssrab 3251* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssrabdv 3252* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)

Theoremrabssdv 3253* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)

Theoremss2rabdv 3254* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)

Theoremss2rabi 3255 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)

Theoremrabss2 3256* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssab2 3257* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)

Theoremssrab2 3258* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)

Theoremrabssab 3259 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremuniiunlem 3260* A subset relationship useful for converting union to indexed union using dfiun2 3937 or dfiun2g 3935 and intersection to indexed intersection using dfiin2 3938. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Theoremdfpss2 3261 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)

Theoremdfpss3 3262 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempsseq1 3263 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)

Theorempsseq2 3264 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)

Theorempsseq1i 3265 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq2i 3266 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq12i 3267 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq1d 3268 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq2d 3269 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq12d 3270 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempssss 3271 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theorempssne 3272 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)

Theorempssssd 3273 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)

Theorempssned 3274 Proper subclasses are unequal. Deduction form of pssne 3272. (Contributed by David Moews, 1-May-2017.)

Theoremsspss 3275 Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)

Theorempssirr 3276 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theorempssn2lp 3277 Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremsspsstri 3278 Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)

Theoremssnpss 3279 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempsstr 3280 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theoremsspsstr 3281 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Theorempsssstr 3282 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Theorempsstrd 3283 Proper subclass inclusion is transitive. Deduction form of psstr 3280. (Contributed by David Moews, 1-May-2017.)

Theoremsspsstrd 3284 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3281. (Contributed by David Moews, 1-May-2017.)

Theorempsssstrd 3285 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3282. (Contributed by David Moews, 1-May-2017.)

Theoremnpss 3286 A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3194. (Contributed by Mario Carneiro, 15-May-2015.)

2.1.13  The difference, union, and intersection of two classes

Theoremdifeq1 3287 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifeq2 3288 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifeq12 3289 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)

Theoremdifeq1i 3290 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2i 3291 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12i 3292 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)

Theoremdifeq1d 3293 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2d 3294 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12d 3295 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)

Theoremdifeqri 3296* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnfdif 3297 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremeldifi 3298 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)

Theoremeldifn 3299 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)

Theoremelndif 3300 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)

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