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Theorem List for New Foundations Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqsstri 3301 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)

Theoremeqsstr3i 3302 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)

Theoremsseqtri 3303 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)

Theoremsseqtr4i 3304 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)

Theoremeqsstrd 3305 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremeqsstr3d 3306 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtrd 3307 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtr4d 3308 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theorem3sstr3i 3309 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 3310 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 3311 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4g 3312 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 3313 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4d 3314 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 3315 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl5eqssr 3316 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl6sseq 3317 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl6sseqr 3318 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl5sseq 3319 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsyl5sseqr 3320 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsyl6eqss 3321 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremsyl6eqssr 3322 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremeqimss 3323 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremeqimss2 3324 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)

Theoremeqimssi 3325 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)

Theoremeqimss2i 3326 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)

Theoremnssne1 3327 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnssne2 3328 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnss 3329* 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 3330* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)

Theoremssrexv 3331* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)

Theoremralss 3332* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremrexss 3333* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremss2ab 3334 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)

Theoremabss 3335* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssab 3336* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssabral 3337* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)

Theoremss2abi 3338 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)

Theoremss2abdv 3339* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)

Theoremabssdv 3340* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremabssi 3341* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremss2rab 3342 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)

Theoremrabss 3343* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssrab 3344* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssrabdv 3345* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)

Theoremrabssdv 3346* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)

Theoremss2rabdv 3347* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)

Theoremss2rabi 3348 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)

Theoremrabss2 3349* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssab2 3350* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)

Theoremssrab2 3351* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)

Theoremrabssab 3352 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremuniiunlem 3353* A subset relationship useful for converting union to indexed union using dfiun2 4001 or dfiun2g 3999 and intersection to indexed intersection using dfiin2 4002. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Theoremdfpss2 3354 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)

Theoremdfpss3 3355 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempsseq1 3356 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)

Theorempsseq2 3357 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)

Theorempsseq1i 3358 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq2i 3359 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq12i 3360 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq1d 3361 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq2d 3362 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq12d 3363 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempssss 3364 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theorempssne 3365 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)

Theorempssssd 3366 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)

Theorempssned 3367 Proper subclasses are unequal. Deduction form of pssne 3365. (Contributed by David Moews, 1-May-2017.)

Theoremsspss 3368 Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)

Theorempssirr 3369 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theorempssn2lp 3370 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 3371 Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)

Theoremssnpss 3372 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempsstr 3373 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theoremsspsstr 3374 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Theorempsssstr 3375 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Theorempsstrd 3376 Proper subclass inclusion is transitive. Deduction form of psstr 3373. (Contributed by David Moews, 1-May-2017.)

Theoremsspsstrd 3377 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3374. (Contributed by David Moews, 1-May-2017.)

Theorempsssstrd 3378 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3375. (Contributed by David Moews, 1-May-2017.)

Theoremnpss 3379 A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3287. (Contributed by Mario Carneiro, 15-May-2015.)

2.1.12  The difference, union, and intersection of two classes

Theoremdifeq12 3380 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)

Theoremdifeq1i 3381 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2i 3382 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12i 3383 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)

Theoremdifeq1d 3384 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2d 3385 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12d 3386 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)

Theoremdifeqri 3387* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremeldifi 3388 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)

Theoremeldifn 3389 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)

Theoremelndif 3390 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)

Theoremneldif 3391 Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)

Theoremdifdif 3392 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)

Theoremdifss 3393 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)

Theoremdifssd 3394 A difference of two classes is contained in the minuend. Deduction form of difss 3393. (Contributed by David Moews, 1-May-2017.)

Theoremdifss2 3395 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)

Theoremdifss2d 3396 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3395. (Contributed by David Moews, 1-May-2017.)

Theoremssdifss 3397 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)

Theoremddif 3398 Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Theoremssconb 3399 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)

Theoremsscon 3400 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)

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