Theorem List for New Foundations Explorer - 3301-3400 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | eqsstri 3301 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 16-Jul-1995.)
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Theorem | eqsstr3i 3302 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 19-Oct-1999.)
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Theorem | sseqtri 3303 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 28-Jul-1995.)
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Theorem | sseqtr4i 3304 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 4-Apr-1995.)
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Theorem | eqsstrd 3305 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | eqsstr3d 3306 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | sseqtrd 3307 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | sseqtr4d 3308 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | 3sstr3i 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.)
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Theorem | 3sstr4i 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.)
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Theorem | 3sstr3g 3311 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000.)
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Theorem | 3sstr4g 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.)
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Theorem | 3sstr3d 3313 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000.)
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Theorem | 3sstr4d 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.)
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Theorem | syl5eqss 3315 |
B chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | syl5eqssr 3316 |
B chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | syl6sseq 3317 |
A chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | syl6sseqr 3318 |
A chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | syl5sseq 3319 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | syl5sseqr 3320 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | syl6eqss 3321 |
A chained subclass and equality deduction. (Contributed by Mario
Carneiro, 2-Jan-2017.)
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Theorem | syl6eqssr 3322 |
A chained subclass and equality deduction. (Contributed by Mario
Carneiro, 2-Jan-2017.)
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Theorem | eqimss 3323 |
Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.)
(Proof shortened by Andrew Salmon, 21-Jun-2011.)
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Theorem | eqimss2 3324 |
Equality implies the subclass relation. (Contributed by NM,
23-Nov-2003.)
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Theorem | eqimssi 3325 |
Infer subclass relationship from equality. (Contributed by NM,
6-Jan-2007.)
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Theorem | eqimss2i 3326 |
Infer subclass relationship from equality. (Contributed by NM,
7-Jan-2007.)
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Theorem | nssne1 3327 |
Two classes are different if they don't include the same class.
(Contributed by NM, 23-Apr-2015.)
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Theorem | nssne2 3328 |
Two classes are different if they are not subclasses of the same class.
(Contributed by NM, 23-Apr-2015.)
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Theorem | nss 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.)
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Theorem | ssralv 3330* |
Quantification restricted to a subclass. (Contributed by NM,
11-Mar-2006.)
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Theorem | ssrexv 3331* |
Existential quantification restricted to a subclass. (Contributed by
NM, 11-Jan-2007.)
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Theorem | ralss 3332* |
Restricted universal quantification on a subset in terms of superset.
(Contributed by Stefan O'Rear, 3-Apr-2015.)
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Theorem | rexss 3333* |
Restricted existential quantification on a subset in terms of superset.
(Contributed by Stefan O'Rear, 3-Apr-2015.)
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Theorem | ss2ab 3334 |
Class abstractions in a subclass relationship. (Contributed by NM,
3-Jul-1994.)
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Theorem | abss 3335* |
Class abstraction in a subclass relationship. (Contributed by NM,
16-Aug-2006.)
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Theorem | ssab 3336* |
Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
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Theorem | ssabral 3337* |
The relation for a subclass of a class abstraction is equivalent to
restricted quantification. (Contributed by NM, 6-Sep-2006.)
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Theorem | ss2abi 3338 |
Inference of abstraction subclass from implication. (Contributed by NM,
31-Mar-1995.)
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Theorem | ss2abdv 3339* |
Deduction of abstraction subclass from implication. (Contributed by NM,
29-Jul-2011.)
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Theorem | abssdv 3340* |
Deduction of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
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Theorem | abssi 3341* |
Inference of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
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Theorem | ss2rab 3342 |
Restricted abstraction classes in a subclass relationship. (Contributed
by NM, 30-May-1999.)
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Theorem | rabss 3343* |
Restricted class abstraction in a subclass relationship. (Contributed
by NM, 16-Aug-2006.)
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Theorem | ssrab 3344* |
Subclass of a restricted class abstraction. (Contributed by NM,
16-Aug-2006.)
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Theorem | ssrabdv 3345* |
Subclass of a restricted class abstraction (deduction rule).
(Contributed by NM, 31-Aug-2006.)
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Theorem | rabssdv 3346* |
Subclass of a restricted class abstraction (deduction rule).
(Contributed by NM, 2-Feb-2015.)
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Theorem | ss2rabdv 3347* |
Deduction of restricted abstraction subclass from implication.
(Contributed by NM, 30-May-2006.)
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Theorem | ss2rabi 3348 |
Inference of restricted abstraction subclass from implication.
(Contributed by NM, 14-Oct-1999.)
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Theorem | rabss2 3349* |
Subclass law for restricted abstraction. (Contributed by NM,
18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | ssab2 3350* |
Subclass relation for the restriction of a class abstraction.
(Contributed by NM, 31-Mar-1995.)
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Theorem | ssrab2 3351* |
Subclass relation for a restricted class. (Contributed by NM,
19-Mar-1997.)
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Theorem | rabssab 3352 |
A restricted class is a subclass of the corresponding unrestricted class.
(Contributed by Mario Carneiro, 23-Dec-2016.)
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Theorem | uniiunlem 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.)
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Theorem | dfpss2 3354 |
Alternate definition of proper subclass. (Contributed by NM,
7-Feb-1996.)
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Theorem | dfpss3 3355 |
Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | psseq1 3356 |
Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
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Theorem | psseq2 3357 |
Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
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Theorem | psseq1i 3358 |
An equality inference for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
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Theorem | psseq2i 3359 |
An equality inference for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
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Theorem | psseq12i 3360 |
An equality inference for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
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Theorem | psseq1d 3361 |
An equality deduction for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
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Theorem | psseq2d 3362 |
An equality deduction for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
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Theorem | psseq12d 3363 |
An equality deduction for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
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Theorem | pssss 3364 |
A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23.
(Contributed by NM, 7-Feb-1996.)
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Theorem | pssne 3365 |
Two classes in a proper subclass relationship are not equal. (Contributed
by NM, 16-Feb-2015.)
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Theorem | pssssd 3366 |
Deduce subclass from proper subclass. (Contributed by NM,
29-Feb-1996.)
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Theorem | pssned 3367 |
Proper subclasses are unequal. Deduction form of pssne 3365.
(Contributed by David Moews, 1-May-2017.)
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Theorem | sspss 3368 |
Subclass in terms of proper subclass. (Contributed by NM,
25-Feb-1996.)
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Theorem | pssirr 3369 |
Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23.
(Contributed by NM, 7-Feb-1996.)
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Theorem | pssn2lp 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.)
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Theorem | sspsstri 3371 |
Two ways of stating trichotomy with respect to inclusion. (Contributed by
NM, 12-Aug-2004.)
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Theorem | ssnpss 3372 |
Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | psstr 3373 |
Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23.
(Contributed by NM, 7-Feb-1996.)
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Theorem | sspsstr 3374 |
Transitive law for subclass and proper subclass. (Contributed by NM,
3-Apr-1996.)
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Theorem | psssstr 3375 |
Transitive law for subclass and proper subclass. (Contributed by NM,
3-Apr-1996.)
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Theorem | psstrd 3376 |
Proper subclass inclusion is transitive. Deduction form of psstr 3373.
(Contributed by David Moews, 1-May-2017.)
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Theorem | sspsstrd 3377 |
Transitivity involving subclass and proper subclass inclusion.
Deduction form of sspsstr 3374. (Contributed by David Moews,
1-May-2017.)
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Theorem | psssstrd 3378 |
Transitivity involving subclass and proper subclass inclusion.
Deduction form of psssstr 3375. (Contributed by David Moews,
1-May-2017.)
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Theorem | npss 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.)
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2.1.12 The difference, union, and intersection
of two classes
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Theorem | difeq12 3380 |
Equality theorem for class difference. (Contributed by FL,
31-Aug-2009.)
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Theorem | difeq1i 3381 |
Inference adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq2i 3382 |
Inference adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq12i 3383 |
Equality inference for class difference. (Contributed by NM,
29-Aug-2004.)
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Theorem | difeq1d 3384 |
Deduction adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq2d 3385 |
Deduction adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq12d 3386 |
Equality deduction for class difference. (Contributed by FL,
29-May-2014.)
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Theorem | difeqri 3387* |
Inference from membership to difference. (Contributed by NM,
17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | eldifi 3388 |
Implication of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
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Theorem | eldifn 3389 |
Implication of membership in a class difference. (Contributed by NM,
3-May-1994.)
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Theorem | elndif 3390 |
A set does not belong to a class excluding it. (Contributed by NM,
27-Jun-1994.)
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Theorem | neldif 3391 |
Implication of membership in a class difference. (Contributed by NM,
28-Jun-1994.)
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Theorem | difdif 3392 |
Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
(Contributed by NM, 17-May-1998.)
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Theorem | difss 3393 |
Subclass relationship for class difference. Exercise 14 of
[TakeutiZaring] p. 22.
(Contributed by NM, 29-Apr-1994.)
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Theorem | difssd 3394 |
A difference of two classes is contained in the minuend. Deduction form
of difss 3393. (Contributed by David Moews, 1-May-2017.)
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Theorem | difss2 3395 |
If a class is contained in a difference, it is contained in the minuend.
(Contributed by David Moews, 1-May-2017.)
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Theorem | difss2d 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.)
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Theorem | ssdifss 3397 |
Preservation of a subclass relationship by class difference. (Contributed
by NM, 15-Feb-2007.)
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Theorem | ddif 3398 |
Double complement under universal class. Exercise 4.10(s) of
[Mendelson] p. 231. (Contributed by
NM, 8-Jan-2002.)
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Theorem | ssconb 3399 |
Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
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Theorem | sscon 3400 |
Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
(Contributed by NM, 22-Mar-1998.)
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