Theorem List for Intuitionistic Logic Explorer - 3301-3400 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | unss 3301 |
The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27
and its converse. (Contributed by NM, 11-Jun-2004.)
|
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) |
|
Theorem | unssi 3302 |
An inference showing the union of two subclasses is a subclass.
(Contributed by Raph Levien, 10-Dec-2002.)
|
⊢ 𝐴 ⊆ 𝐶
& ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ (𝐴 ∪ 𝐵) ⊆ 𝐶 |
|
Theorem | unssd 3303 |
A deduction showing the union of two subclasses is a subclass.
(Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐶)
& ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
|
Theorem | unssad 3304 |
If (𝐴
∪ 𝐵) is
contained in 𝐶, so is 𝐴. One-way
deduction form of unss 3301. Partial converse of unssd 3303. (Contributed
by David Moews, 1-May-2017.)
|
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
|
Theorem | unssbd 3305 |
If (𝐴
∪ 𝐵) is
contained in 𝐶, so is 𝐵. One-way
deduction form of unss 3301. Partial converse of unssd 3303. (Contributed
by David Moews, 1-May-2017.)
|
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
|
Theorem | ssun 3306 |
A condition that implies inclusion in the union of two classes.
(Contributed by NM, 23-Nov-2003.)
|
⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
|
Theorem | rexun 3307 |
Restricted existential quantification over union. (Contributed by Jeff
Madsen, 5-Jan-2011.)
|
⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑)) |
|
Theorem | ralunb 3308 |
Restricted quantification over a union. (Contributed by Scott Fenton,
12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|
⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑)) |
|
Theorem | ralun 3309 |
Restricted quantification over union. (Contributed by Jeff Madsen,
2-Sep-2009.)
|
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑) |
|
2.1.13.3 The intersection of two
classes
|
|
Theorem | elin 3310 |
Expansion of membership in an intersection of two classes. Theorem 12
of [Suppes] p. 25. (Contributed by NM,
29-Apr-1994.)
|
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
|
Theorem | elini 3311 |
Membership in an intersection of two classes. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
⊢ 𝐴 ∈ 𝐵
& ⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) |
|
Theorem | elind 3312 |
Deduce membership in an intersection of two classes. (Contributed by
Jonathan Ben-Naim, 3-Jun-2011.)
|
⊢ (𝜑 → 𝑋 ∈ 𝐴)
& ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
|
Theorem | elinel1 3313 |
Membership in an intersection implies membership in the first set.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵) |
|
Theorem | elinel2 3314 |
Membership in an intersection implies membership in the second set.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐶) |
|
Theorem | elin2 3315 |
Membership in a class defined as an intersection. (Contributed by
Stefan O'Rear, 29-Mar-2015.)
|
⊢ 𝑋 = (𝐵 ∩ 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
|
Theorem | elin1d 3316 |
Elementhood in the first set of an intersection - deduction version.
(Contributed by Thierry Arnoux, 3-May-2020.)
|
⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
|
Theorem | elin2d 3317 |
Elementhood in the first set of an intersection - deduction version.
(Contributed by Thierry Arnoux, 3-May-2020.)
|
⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
|
Theorem | elin3 3318 |
Membership in a class defined as a ternary intersection. (Contributed
by Stefan O'Rear, 29-Mar-2015.)
|
⊢ 𝑋 = ((𝐵 ∩ 𝐶) ∩ 𝐷) ⇒ ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶 ∧ 𝐴 ∈ 𝐷)) |
|
Theorem | incom 3319 |
Commutative law for intersection of classes. Exercise 7 of
[TakeutiZaring] p. 17.
(Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) |
|
Theorem | ineqri 3320* |
Inference from membership to intersection. (Contributed by NM,
5-Aug-1993.)
|
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
|
Theorem | ineq1 3321 |
Equality theorem for intersection of two classes. (Contributed by NM,
14-Dec-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
|
Theorem | ineq2 3322 |
Equality theorem for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
|
Theorem | ineq12 3323 |
Equality theorem for intersection of two classes. (Contributed by NM,
8-May-1994.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
|
Theorem | ineq1i 3324 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) |
|
Theorem | ineq2i 3325 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵) |
|
Theorem | ineq12i 3326 |
Equality inference for intersection of two classes. (Contributed by
NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|
⊢ 𝐴 = 𝐵
& ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷) |
|
Theorem | ineq1d 3327 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
|
Theorem | ineq2d 3328 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
|
Theorem | ineq12d 3329 |
Equality deduction for intersection of two classes. (Contributed by
NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
|
Theorem | ineqan12d 3330 |
Equality deduction for intersection of two classes. (Contributed by
NM, 7-Feb-2007.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜓 → 𝐶 = 𝐷) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
|
Theorem | dfss1 3331 |
A frequently-used variant of subclass definition df-ss 3134. (Contributed
by NM, 10-Jan-2015.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
|
Theorem | dfss5 3332 |
Another definition of subclasshood. Similar to df-ss 3134, dfss 3135, and
dfss1 3331. (Contributed by David Moews, 1-May-2017.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴)) |
|
Theorem | nfin 3333 |
Bound-variable hypothesis builder for the intersection of classes.
(Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro,
14-Oct-2016.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) |
|
Theorem | csbing 3334 |
Distribute proper substitution through an intersection relation.
(Contributed by Alan Sare, 22-Jul-2012.)
|
⊢ (𝐴 ∈ 𝐵 → ⦋𝐴 / 𝑥⦌(𝐶 ∩ 𝐷) = (⦋𝐴 / 𝑥⦌𝐶 ∩ ⦋𝐴 / 𝑥⦌𝐷)) |
|
Theorem | rabbi2dva 3335* |
Deduction from a wff to a restricted class abstraction. (Contributed by
NM, 14-Jan-2014.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝜓)) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝜓}) |
|
Theorem | inidm 3336 |
Idempotent law for intersection of classes. Theorem 15 of [Suppes]
p. 26. (Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∩ 𝐴) = 𝐴 |
|
Theorem | inass 3337 |
Associative law for intersection of classes. Exercise 9 of
[TakeutiZaring] p. 17.
(Contributed by NM, 3-May-1994.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) |
|
Theorem | in12 3338 |
A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
|
Theorem | in32 3339 |
A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵) |
|
Theorem | in13 3340 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴)) |
|
Theorem | in31 3341 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴) |
|
Theorem | inrot 3342 |
Rotate the intersection of 3 classes. (Contributed by NM,
27-Aug-2012.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵) |
|
Theorem | in4 3343 |
Rearrangement of intersection of 4 classes. (Contributed by NM,
21-Apr-2001.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) |
|
Theorem | inindi 3344 |
Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) |
|
Theorem | inindir 3345 |
Intersection distributes over itself. (Contributed by NM,
17-Aug-2004.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) |
|
Theorem | sseqin2 3346 |
A relationship between subclass and intersection. Similar to Exercise 9
of [TakeutiZaring] p. 18.
(Contributed by NM, 17-May-1994.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
|
Theorem | inss1 3347 |
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.)
|
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 |
|
Theorem | inss2 3348 |
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.)
|
⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 |
|
Theorem | ssin 3349 |
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.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) |
|
Theorem | ssini 3350 |
An inference showing that a subclass of two classes is a subclass of
their intersection. (Contributed by NM, 24-Nov-2003.)
|
⊢ 𝐴 ⊆ 𝐵
& ⊢ 𝐴 ⊆ 𝐶 ⇒ ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶) |
|
Theorem | ssind 3351 |
A deduction showing that a subclass of two classes is a subclass of
their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵)
& ⊢ (𝜑 → 𝐴 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∩ 𝐶)) |
|
Theorem | ssrin 3352 |
Add right intersection to subclass relation. (Contributed by NM,
16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) |
|
Theorem | sslin 3353 |
Add left intersection to subclass relation. (Contributed by NM,
19-Oct-1999.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵)) |
|
Theorem | ssrind 3354 |
Add right intersection to subclass relation. (Contributed by Glauco
Siliprandi, 2-Jan-2022.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) |
|
Theorem | ss2in 3355 |
Intersection of subclasses. (Contributed by NM, 5-May-2000.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷)) |
|
Theorem | ssinss1 3356 |
Intersection preserves subclass relationship. (Contributed by NM,
14-Sep-1999.)
|
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
|
Theorem | inss 3357 |
Inclusion of an intersection of two classes. (Contributed by NM,
30-Oct-2014.)
|
⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
|
2.1.13.4 Combinations of difference, union, and
intersection of two classes
|
|
Theorem | unabs 3358 |
Absorption law for union. (Contributed by NM, 16-Apr-2006.)
|
⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴 |
|
Theorem | inabs 3359 |
Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
|
⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴 |
|
Theorem | dfss4st 3360* |
Subclass defined in terms of class difference. (Contributed by NM,
22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (∀𝑥STAB 𝑥 ∈ 𝐴 → (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴)) |
|
Theorem | ssddif 3361 |
Double complement and subset. Similar to ddifss 3365 but inside a class
𝐵 instead of the universal class V. In classical logic the
subset operation on the right hand side could be an equality (that is,
𝐴
⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴). (Contributed by Jim Kingdon,
24-Jul-2018.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ (𝐵 ∖ (𝐵 ∖ 𝐴))) |
|
Theorem | unssdif 3362 |
Union of two classes and class difference. In classical logic this
would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
|
⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) |
|
Theorem | inssdif 3363 |
Intersection of two classes and class difference. In classical logic
this would be an equality. (Contributed by Jim Kingdon,
24-Jul-2018.)
|
⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵)) |
|
Theorem | difin 3364 |
Difference with intersection. Theorem 33 of [Suppes] p. 29.
(Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon,
26-Jun-2011.)
|
⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) |
|
Theorem | ddifss 3365 |
Double complement under universal class. In classical logic (or given an
additional hypothesis, as in ddifnel 3258), this is equality rather than
subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
|
⊢ 𝐴 ⊆ (V ∖ (V ∖ 𝐴)) |
|
Theorem | unssin 3366 |
Union as a subset of class complement and intersection (De Morgan's
law). One direction of the definition of union in [Mendelson] p. 231.
This would be an equality, rather than subset, in classical logic.
(Contributed by Jim Kingdon, 25-Jul-2018.)
|
⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
|
Theorem | inssun 3367 |
Intersection in terms of class difference and union (De Morgan's law).
Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an
equality, rather than subset, in classical logic. (Contributed by Jim
Kingdon, 25-Jul-2018.)
|
⊢ (𝐴 ∩ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) |
|
Theorem | inssddif 3368 |
Intersection of two classes and class difference. In classical logic,
such as Exercise 4.10(q) of [Mendelson]
p. 231, this is an equality rather
than subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
|
⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∖ (𝐴 ∖ 𝐵)) |
|
Theorem | invdif 3369 |
Intersection with universal complement. Remark in [Stoll] p. 20.
(Contributed by NM, 17-Aug-2004.)
|
⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
|
Theorem | indif 3370 |
Intersection with class difference. Theorem 34 of [Suppes] p. 29.
(Contributed by NM, 17-Aug-2004.)
|
⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
|
Theorem | indif2 3371 |
Bring an intersection in and out of a class difference. (Contributed by
Jeff Hankins, 15-Jul-2009.)
|
⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
|
Theorem | indif1 3372 |
Bring an intersection in and out of a class difference. (Contributed by
Mario Carneiro, 15-May-2015.)
|
⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
|
Theorem | indifcom 3373 |
Commutation law for intersection and difference. (Contributed by Scott
Fenton, 18-Feb-2013.)
|
⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶)) |
|
Theorem | indi 3374 |
Distributive law for intersection over union. Exercise 10 of
[TakeutiZaring] p. 17.
(Contributed by NM, 30-Sep-2002.) (Proof
shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (𝐴 ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)) |
|
Theorem | undi 3375 |
Distributive law for union over intersection. Exercise 11 of
[TakeutiZaring] p. 17.
(Contributed by NM, 30-Sep-2002.) (Proof
shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (𝐴 ∪ (𝐵 ∩ 𝐶)) = ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)) |
|
Theorem | indir 3376 |
Distributive law for intersection over union. Theorem 28 of [Suppes]
p. 27. (Contributed by NM, 30-Sep-2002.)
|
⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) |
|
Theorem | undir 3377 |
Distributive law for union over intersection. Theorem 29 of [Suppes]
p. 27. (Contributed by NM, 30-Sep-2002.)
|
⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)) |
|
Theorem | uneqin 3378 |
Equality of union and intersection implies equality of their arguments.
(Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon,
26-Jun-2011.)
|
⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) ↔ 𝐴 = 𝐵) |
|
Theorem | difundi 3379 |
Distributive law for class difference. Theorem 39 of [Suppes] p. 29.
(Contributed by NM, 17-Aug-2004.)
|
⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) |
|
Theorem | difundir 3380 |
Distributive law for class difference. (Contributed by NM,
17-Aug-2004.)
|
⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶)) |
|
Theorem | difindiss 3381 |
Distributive law for class difference. In classical logic, for example,
theorem 40 of [Suppes] p. 29, this is an
equality instead of subset.
(Contributed by Jim Kingdon, 26-Jul-2018.)
|
⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ 𝐶)) |
|
Theorem | difindir 3382 |
Distributive law for class difference. (Contributed by NM,
17-Aug-2004.)
|
⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (𝐵 ∖ 𝐶)) |
|
Theorem | indifdir 3383 |
Distribute intersection over difference. (Contributed by Scott Fenton,
14-Apr-2011.)
|
⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶)) |
|
Theorem | difdif2ss 3384 |
Set difference with a set difference. In classical logic this would be
equality rather than subset. (Contributed by Jim Kingdon,
27-Jul-2018.)
|
⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∖ 𝐶)) |
|
Theorem | undm 3385 |
De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19.
(Contributed by NM, 18-Aug-2004.)
|
⊢ (V ∖ (𝐴 ∪ 𝐵)) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) |
|
Theorem | indmss 3386 |
De Morgan's law for intersection. In classical logic, this would be
equality rather than subset, as in Theorem 5.2(13') of [Stoll] p. 19.
(Contributed by Jim Kingdon, 27-Jul-2018.)
|
⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) ⊆ (V ∖ (𝐴 ∩ 𝐵)) |
|
Theorem | difun1 3387 |
A relationship involving double difference and union. (Contributed by NM,
29-Aug-2004.)
|
⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶) |
|
Theorem | undif3ss 3388 |
A subset relationship involving class union and class difference. In
classical logic, this would be equality rather than subset, as in the
first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by
Jim Kingdon, 28-Jul-2018.)
|
⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) ⊆ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) |
|
Theorem | difin2 3389 |
Represent a set difference as an intersection with a larger difference.
(Contributed by Jeff Madsen, 2-Sep-2009.)
|
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∖ 𝐵) = ((𝐶 ∖ 𝐵) ∩ 𝐴)) |
|
Theorem | dif32 3390 |
Swap second and third argument of double difference. (Contributed by NM,
18-Aug-2004.)
|
⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵) |
|
Theorem | difabs 3391 |
Absorption-like law for class difference: you can remove a class only
once. (Contributed by FL, 2-Aug-2009.)
|
⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) |
|
Theorem | symdif1 3392 |
Two ways to express symmetric difference. This theorem shows the
equivalence of the definition of symmetric difference in [Stoll] p. 13 and
the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by
NM, 17-Aug-2004.)
|
⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) |
|
2.1.13.5 Class abstractions with difference,
union, and intersection of two classes
|
|
Theorem | symdifxor 3393* |
Expressing symmetric difference with exclusive-or or two differences.
(Contributed by Jim Kingdon, 28-Jul-2018.)
|
⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} |
|
Theorem | unab 3394 |
Union of two class abstractions. (Contributed by NM, 29-Sep-2002.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∨ 𝜓)} |
|
Theorem | inab 3395 |
Intersection of two class abstractions. (Contributed by NM,
29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ ({𝑥 ∣ 𝜑} ∩ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∧ 𝜓)} |
|
Theorem | difab 3396 |
Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ ({𝑥 ∣ 𝜑} ∖ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)} |
|
Theorem | notab 3397 |
A class builder defined by a negation. (Contributed by FL,
18-Sep-2010.)
|
⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑}) |
|
Theorem | unrab 3398 |
Union of two restricted class abstractions. (Contributed by NM,
25-Mar-2004.)
|
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)} |
|
Theorem | inrab 3399 |
Intersection of two restricted class abstractions. (Contributed by NM,
1-Sep-2006.)
|
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} |
|
Theorem | inrab2 3400* |
Intersection with a restricted class abstraction. (Contributed by NM,
19-Nov-2007.)
|
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ 𝜑} |