Theorem List for Intuitionistic Logic Explorer - 3201-3300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | uneq12d 3201 |
Equality deduction for union of two classes. (Contributed by NM,
29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
|
Theorem | nfun 3202 |
Bound-variable hypothesis builder for the union of classes.
(Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro,
14-Oct-2016.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵) |
|
Theorem | unass 3203 |
Associative law for union of classes. Exercise 8 of [TakeutiZaring]
p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew
Salmon, 26-Jun-2011.)
|
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) |
|
Theorem | un12 3204 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
|
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) |
|
Theorem | un23 3205 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof
shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ 𝐵) |
|
Theorem | un4 3206 |
A rearrangement of the union of 4 classes. (Contributed by NM,
12-Aug-2004.)
|
⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) |
|
Theorem | unundi 3207 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) |
|
Theorem | unundir 3208 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶)) |
|
Theorem | ssun1 3209 |
Subclass relationship for union of classes. Theorem 25 of [Suppes]
p. 27. (Contributed by NM, 5-Aug-1993.)
|
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) |
|
Theorem | ssun2 3210 |
Subclass relationship for union of classes. (Contributed by NM,
30-Aug-1993.)
|
⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) |
|
Theorem | ssun3 3211 |
Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
|
Theorem | ssun4 3212 |
Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
|
Theorem | elun1 3213 |
Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶)) |
|
Theorem | elun2 3214 |
Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
|
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵)) |
|
Theorem | unss1 3215 |
Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) |
|
Theorem | ssequn1 3216 |
A relationship between subclass and union. Theorem 26 of [Suppes]
p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew
Salmon, 26-Jun-2011.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵) |
|
Theorem | unss2 3217 |
Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18.
(Contributed by NM, 14-Oct-1999.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∪ 𝐴) ⊆ (𝐶 ∪ 𝐵)) |
|
Theorem | unss12 3218 |
Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) |
|
Theorem | ssequn2 3219 |
A relationship between subclass and union. (Contributed by NM,
13-Jun-1994.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) |
|
Theorem | unss 3220 |
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 3221 |
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 3222 |
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 3223 |
If (𝐴
∪ 𝐵) is
contained in 𝐶, so is 𝐴. One-way
deduction form of unss 3220. Partial converse of unssd 3222. (Contributed
by David Moews, 1-May-2017.)
|
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
|
Theorem | unssbd 3224 |
If (𝐴
∪ 𝐵) is
contained in 𝐶, so is 𝐵. One-way
deduction form of unss 3220. Partial converse of unssd 3222. (Contributed
by David Moews, 1-May-2017.)
|
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
|
Theorem | ssun 3225 |
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 3226 |
Restricted existential quantification over union. (Contributed by Jeff
Madsen, 5-Jan-2011.)
|
⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑)) |
|
Theorem | ralunb 3227 |
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 3228 |
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 3229 |
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 3230 |
Membership in an intersection of two classes. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
⊢ 𝐴 ∈ 𝐵
& ⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) |
|
Theorem | elind 3231 |
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 3232 |
Membership in an intersection implies membership in the first set.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵) |
|
Theorem | elinel2 3233 |
Membership in an intersection implies membership in the second set.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐶) |
|
Theorem | elin2 3234 |
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 3235 |
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 3236 |
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 3237 |
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 3238 |
Commutative law for intersection of classes. Exercise 7 of
[TakeutiZaring] p. 17.
(Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) |
|
Theorem | ineqri 3239* |
Inference from membership to intersection. (Contributed by NM,
5-Aug-1993.)
|
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
|
Theorem | ineq1 3240 |
Equality theorem for intersection of two classes. (Contributed by NM,
14-Dec-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
|
Theorem | ineq2 3241 |
Equality theorem for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
|
Theorem | ineq12 3242 |
Equality theorem for intersection of two classes. (Contributed by NM,
8-May-1994.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
|
Theorem | ineq1i 3243 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) |
|
Theorem | ineq2i 3244 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵) |
|
Theorem | ineq12i 3245 |
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 3246 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
|
Theorem | ineq2d 3247 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
|
Theorem | ineq12d 3248 |
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 3249 |
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 3250 |
A frequently-used variant of subclass definition df-ss 3054. (Contributed
by NM, 10-Jan-2015.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
|
Theorem | dfss5 3251 |
Another definition of subclasshood. Similar to df-ss 3054, dfss 3055, and
dfss1 3250. (Contributed by David Moews, 1-May-2017.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴)) |
|
Theorem | nfin 3252 |
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 3253 |
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 3254* |
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 3255 |
Idempotent law for intersection of classes. Theorem 15 of [Suppes]
p. 26. (Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∩ 𝐴) = 𝐴 |
|
Theorem | inass 3256 |
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 3257 |
A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
|
Theorem | in32 3258 |
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 3259 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴)) |
|
Theorem | in31 3260 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴) |
|
Theorem | inrot 3261 |
Rotate the intersection of 3 classes. (Contributed by NM,
27-Aug-2012.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵) |
|
Theorem | in4 3262 |
Rearrangement of intersection of 4 classes. (Contributed by NM,
21-Apr-2001.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) |
|
Theorem | inindi 3263 |
Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) |
|
Theorem | inindir 3264 |
Intersection distributes over itself. (Contributed by NM,
17-Aug-2004.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) |
|
Theorem | sseqin2 3265 |
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 3266 |
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 3267 |
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 3268 |
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 3269 |
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 3270 |
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 3271 |
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 3272 |
Add left intersection to subclass relation. (Contributed by NM,
19-Oct-1999.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵)) |
|
Theorem | ssrind 3273 |
Add right intersection to subclass relation. (Contributed by Glauco
Siliprandi, 2-Jan-2022.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) |
|
Theorem | ss2in 3274 |
Intersection of subclasses. (Contributed by NM, 5-May-2000.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷)) |
|
Theorem | ssinss1 3275 |
Intersection preserves subclass relationship. (Contributed by NM,
14-Sep-1999.)
|
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
|
Theorem | inss 3276 |
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 3277 |
Absorption law for union. (Contributed by NM, 16-Apr-2006.)
|
⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴 |
|
Theorem | inabs 3278 |
Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
|
⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴 |
|
Theorem | dfss4st 3279* |
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 3280 |
Double complement and subset. Similar to ddifss 3284 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 3281 |
Union of two classes and class difference. In classical logic this
would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
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⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) |
|
Theorem | inssdif 3282 |
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 3283 |
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 3284 |
Double complement under universal class. In classical logic (or given an
additional hypothesis, as in ddifnel 3177), this is equality rather than
subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
|
⊢ 𝐴 ⊆ (V ∖ (V ∖ 𝐴)) |
|
Theorem | unssin 3285 |
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 3286 |
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 3287 |
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 3288 |
Intersection with universal complement. Remark in [Stoll] p. 20.
(Contributed by NM, 17-Aug-2004.)
|
⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
|
Theorem | indif 3289 |
Intersection with class difference. Theorem 34 of [Suppes] p. 29.
(Contributed by NM, 17-Aug-2004.)
|
⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
|
Theorem | indif2 3290 |
Bring an intersection in and out of a class difference. (Contributed by
Jeff Hankins, 15-Jul-2009.)
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⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
|
Theorem | indif1 3291 |
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 3292 |
Commutation law for intersection and difference. (Contributed by Scott
Fenton, 18-Feb-2013.)
|
⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶)) |
|
Theorem | indi 3293 |
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 3294 |
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 3295 |
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 3296 |
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 3297 |
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 3298 |
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 3299 |
Distributive law for class difference. (Contributed by NM,
17-Aug-2004.)
|
⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶)) |
|
Theorem | difindiss 3300 |
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)) |