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.)
A = B    &   B C       A C

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

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

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

Theoremeqsstrd 3305 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φA = B)    &   (φB C)       (φA C)

Theoremeqsstr3d 3306 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φB = A)    &   (φB C)       (φA C)

Theoremsseqtrd 3307 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φA B)    &   (φB = C)       (φA C)

Theoremsseqtr4d 3308 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(φA B)    &   (φC = B)       (φA C)

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.)
A B    &   A = C    &   B = D       C D

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.)
A B    &   C = A    &   D = B       C D

Theorem3sstr3g 3311 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(φA B)    &   A = C    &   B = D       (φC D)

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.)
(φA B)    &   C = A    &   D = B       (φC D)

Theorem3sstr3d 3313 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(φA B)    &   (φA = C)    &   (φB = D)       (φC D)

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.)
(φA B)    &   (φC = A)    &   (φD = B)       (φC D)

Theoremsyl5eqss 3315 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
A = B    &   (φB C)       (φA C)

Theoremsyl5eqssr 3316 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
B = A    &   (φB C)       (φA C)

Theoremsyl6sseq 3317 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(φA B)    &   B = C       (φA C)

Theoremsyl6sseqr 3318 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(φA B)    &   C = B       (φA C)

Theoremsyl5sseq 3319 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
B A    &   (φA = C)       (φB C)

Theoremsyl5sseqr 3320 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
B A    &   (φC = A)       (φB C)

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

Theoremsyl6eqssr 3322 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(φB = A)    &   B C       (φA C)

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

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

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

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

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

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

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.)
A Bx(x A ¬ x B))

Theoremssralv 3330* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
(A B → (x B φx A φ))

Theoremssrexv 3331* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
(A B → (x A φx B φ))

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

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

Theoremss2ab 3334 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
({x φ} {x ψ} ↔ x(φψ))

Theoremabss 3335* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({x φ} Ax(φx A))

Theoremssab 3336* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
(A {x φ} ↔ x(x Aφ))

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

Theoremss2abi 3338 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
(φψ)       {x φ} {x ψ}

Theoremss2abdv 3339* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
(φ → (ψχ))       (φ → {x ψ} {x χ})

Theoremabssdv 3340* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(φ → (ψx A))       (φ → {x ψ} A)

Theoremabssi 3341* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(φx A)       {x φ} A

Theoremss2rab 3342 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
({x A φ} {x A ψ} ↔ x A (φψ))

Theoremrabss 3343* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({x A φ} Bx A (φx B))

Theoremssrab 3344* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
(B {x A φ} ↔ (B A x B φ))

Theoremssrabdv 3345* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
(φB A)    &   ((φ x B) → ψ)       (φB {x A ψ})

Theoremrabssdv 3346* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
((φ x A ψ) → x B)       (φ → {x A ψ} B)

Theoremss2rabdv 3347* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
((φ x A) → (ψχ))       (φ → {x A ψ} {x A χ})

Theoremss2rabi 3348 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
(x A → (φψ))       {x A φ} {x A ψ}

Theoremrabss2 3349* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(A B → {x A φ} {x B φ})

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

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

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

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.)
(x A B D → (x A B C ↔ {y x A y = B} C))

Theoremdfpss2 3354 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
(AB ↔ (A B ¬ A = B))

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

Theorempsseq1 3356 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
(A = B → (ACBC))

Theorempsseq2 3357 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
(A = B → (CACB))

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

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

Theorempsseq12i 3360 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
A = B    &   C = D       (ACBD)

Theorempsseq1d 3361 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(φA = B)       (φ → (ACBC))

Theorempsseq2d 3362 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(φA = B)       (φ → (CACB))

Theorempsseq12d 3363 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(φA = B)    &   (φC = D)       (φ → (ACBD))

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

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

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

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

Theoremsspss 3368 Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
(A B ↔ (AB A = B))

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

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.)
¬ (AB BA)

Theoremsspsstri 3371 Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)
((A B B A) ↔ (AB A = B BA))

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

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

Theoremsspsstr 3374 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((A B BC) → AC)

Theorempsssstr 3375 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((AB B C) → AC)

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

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

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

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.)
AB ↔ (A BA = B))

2.1.12  The difference, union, and intersection of two classes

Theoremdifeq12 3380 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
((A = B C = D) → (A C) = (B D))

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

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

Theoremdifeq12i 3383 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
A = B    &   C = D       (A C) = (B D)

Theoremdifeq1d 3384 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
(φA = B)       (φ → (A C) = (B C))

Theoremdifeq2d 3385 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
(φA = B)       (φ → (C A) = (C B))

Theoremdifeq12d 3386 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
(φA = B)    &   (φC = D)       (φ → (A C) = (B D))

Theoremdifeqri 3387* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((x A ¬ x B) ↔ x C)       (A B) = C

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

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

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

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

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

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

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

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

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.)
(φA (B C))       (φA B)

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

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

Theoremssconb 3399 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
((A C B C) → (A (C B) ↔ B (C A)))

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

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