Theorem List for Intuitionistic Logic Explorer - 3201-3300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | difeqri 3201* |
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 |
|
Theorem | nfdif 3202 |
Bound-variable hypothesis builder for class difference. (Contributed by
NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
|
⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵) |
|
Theorem | eldifi 3203 |
Implication of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
|
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵) |
|
Theorem | eldifn 3204 |
Implication of membership in a class difference. (Contributed by NM,
3-May-1994.)
|
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → ¬ 𝐴 ∈ 𝐶) |
|
Theorem | elndif 3205 |
A set does not belong to a class excluding it. (Contributed by NM,
27-Jun-1994.)
|
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) |
|
Theorem | difdif 3206 |
Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
(Contributed by NM, 17-May-1998.)
|
⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴 |
|
Theorem | difss 3207 |
Subclass relationship for class difference. Exercise 14 of
[TakeutiZaring] p. 22.
(Contributed by NM, 29-Apr-1994.)
|
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 |
|
Theorem | difssd 3208 |
A difference of two classes is contained in the minuend. Deduction form
of difss 3207. (Contributed by David Moews, 1-May-2017.)
|
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) |
|
Theorem | difss2 3209 |
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) |
|
Theorem | difss2d 3210 |
If a class is contained in a difference, it is contained in the minuend.
Deduction form of difss2 3209. (Contributed by David Moews,
1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
|
Theorem | ssdifss 3211 |
Preservation of a subclass relationship by class difference. (Contributed
by NM, 15-Feb-2007.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵) |
|
Theorem | ddifnel 3212* |
Double complement under universal class. The hypothesis corresponds to
stability of membership in 𝐴, which is weaker than decidability
(see dcstab 830). Actually, the conclusion is a
characterization of
stability of membership in a class (see ddifstab 3213) . Exercise 4.10(s)
of [Mendelson] p. 231, but with an
additional hypothesis. For a version
without a hypothesis, but which only states that 𝐴 is a subset of
V ∖ (V ∖ 𝐴), see ddifss 3319. (Contributed by Jim Kingdon,
21-Jul-2018.)
|
⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) → 𝑥 ∈ 𝐴) ⇒ ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 |
|
Theorem | ddifstab 3213* |
A class is equal to its double complement if and only if it is stable
(that is, membership in it is a stable property). (Contributed by BJ,
12-Dec-2021.)
|
⊢ ((V ∖ (V ∖ 𝐴)) = 𝐴 ↔ ∀𝑥STAB 𝑥 ∈ 𝐴) |
|
Theorem | ssconb 3214 |
Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
|
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴))) |
|
Theorem | sscon 3215 |
Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
(Contributed by NM, 22-Mar-1998.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
|
Theorem | ssdif 3216 |
Difference law for subsets. (Contributed by NM, 28-May-1998.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
|
Theorem | ssdifd 3217 |
If 𝐴 is contained in 𝐵, then
(𝐴 ∖
𝐶) is contained in
(𝐵
∖ 𝐶).
Deduction form of ssdif 3216. (Contributed by David
Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
|
Theorem | sscond 3218 |
If 𝐴 is contained in 𝐵, then
(𝐶 ∖
𝐵) is contained in
(𝐶
∖ 𝐴).
Deduction form of sscon 3215. (Contributed by David
Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
|
Theorem | ssdifssd 3219 |
If 𝐴 is contained in 𝐵, then
(𝐴 ∖
𝐶) is also contained
in
𝐵. Deduction form of ssdifss 3211. (Contributed by David Moews,
1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵) |
|
Theorem | ssdif2d 3220 |
If 𝐴 is contained in 𝐵 and
𝐶
is contained in 𝐷, then
(𝐴
∖ 𝐷) is
contained in (𝐵 ∖ 𝐶). Deduction form.
(Contributed by David Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵)
& ⊢ (𝜑 → 𝐶 ⊆ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶)) |
|
Theorem | raldifb 3221 |
Restricted universal quantification on a class difference in terms of an
implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
|
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑) |
|
2.1.13.2 The union of two classes
|
|
Theorem | elun 3222 |
Expansion of membership in class union. Theorem 12 of [Suppes] p. 25.
(Contributed by NM, 7-Aug-1994.)
|
⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) |
|
Theorem | uneqri 3223* |
Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
|
⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∪ 𝐵) = 𝐶 |
|
Theorem | unidm 3224 |
Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
(Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∪ 𝐴) = 𝐴 |
|
Theorem | uncom 3225 |
Commutative law for union of classes. Exercise 6 of [TakeutiZaring]
p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew
Salmon, 26-Jun-2011.)
|
⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴) |
|
Theorem | equncom 3226 |
If a class equals the union of two other classes, then it equals the union
of those two classes commuted. (Contributed by Alan Sare,
18-Feb-2012.)
|
⊢ (𝐴 = (𝐵 ∪ 𝐶) ↔ 𝐴 = (𝐶 ∪ 𝐵)) |
|
Theorem | equncomi 3227 |
Inference form of equncom 3226. (Contributed by Alan Sare,
18-Feb-2012.)
|
⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ 𝐴 = (𝐶 ∪ 𝐵) |
|
Theorem | uneq1 3228 |
Equality theorem for union of two classes. (Contributed by NM,
5-Aug-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
|
Theorem | uneq2 3229 |
Equality theorem for the union of two classes. (Contributed by NM,
5-Aug-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) |
|
Theorem | uneq12 3230 |
Equality theorem for union of two classes. (Contributed by NM,
29-Mar-1998.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
|
Theorem | uneq1i 3231 |
Inference adding union to the right in a class equality. (Contributed
by NM, 30-Aug-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) |
|
Theorem | uneq2i 3232 |
Inference adding union to the left in a class equality. (Contributed by
NM, 30-Aug-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵) |
|
Theorem | uneq12i 3233 |
Equality inference for union of two classes. (Contributed by NM,
12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
|
⊢ 𝐴 = 𝐵
& ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷) |
|
Theorem | uneq1d 3234 |
Deduction adding union to the right in a class equality. (Contributed
by NM, 29-Mar-1998.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
|
Theorem | uneq2d 3235 |
Deduction adding union to the left in a class equality. (Contributed by
NM, 29-Mar-1998.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) |
|
Theorem | uneq12d 3236 |
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 3237 |
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 3238 |
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 3239 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
|
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) |
|
Theorem | un23 3240 |
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 3241 |
A rearrangement of the union of 4 classes. (Contributed by NM,
12-Aug-2004.)
|
⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) |
|
Theorem | unundi 3242 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) |
|
Theorem | unundir 3243 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶)) |
|
Theorem | ssun1 3244 |
Subclass relationship for union of classes. Theorem 25 of [Suppes]
p. 27. (Contributed by NM, 5-Aug-1993.)
|
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) |
|
Theorem | ssun2 3245 |
Subclass relationship for union of classes. (Contributed by NM,
30-Aug-1993.)
|
⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) |
|
Theorem | ssun3 3246 |
Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
|
Theorem | ssun4 3247 |
Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
|
Theorem | elun1 3248 |
Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶)) |
|
Theorem | elun2 3249 |
Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
|
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵)) |
|
Theorem | unss1 3250 |
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 3251 |
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 3252 |
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 3253 |
Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) |
|
Theorem | ssequn2 3254 |
A relationship between subclass and union. (Contributed by NM,
13-Jun-1994.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) |
|
Theorem | unss 3255 |
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 3256 |
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 3257 |
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 3258 |
If (𝐴
∪ 𝐵) is
contained in 𝐶, so is 𝐴. One-way
deduction form of unss 3255. Partial converse of unssd 3257. (Contributed
by David Moews, 1-May-2017.)
|
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
|
Theorem | unssbd 3259 |
If (𝐴
∪ 𝐵) is
contained in 𝐶, so is 𝐵. One-way
deduction form of unss 3255. Partial converse of unssd 3257. (Contributed
by David Moews, 1-May-2017.)
|
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
|
Theorem | ssun 3260 |
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 3261 |
Restricted existential quantification over union. (Contributed by Jeff
Madsen, 5-Jan-2011.)
|
⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑)) |
|
Theorem | ralunb 3262 |
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 3263 |
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 3264 |
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 3265 |
Membership in an intersection of two classes. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
⊢ 𝐴 ∈ 𝐵
& ⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) |
|
Theorem | elind 3266 |
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 3267 |
Membership in an intersection implies membership in the first set.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵) |
|
Theorem | elinel2 3268 |
Membership in an intersection implies membership in the second set.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐶) |
|
Theorem | elin2 3269 |
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 3270 |
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 3271 |
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 3272 |
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 3273 |
Commutative law for intersection of classes. Exercise 7 of
[TakeutiZaring] p. 17.
(Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) |
|
Theorem | ineqri 3274* |
Inference from membership to intersection. (Contributed by NM,
5-Aug-1993.)
|
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
|
Theorem | ineq1 3275 |
Equality theorem for intersection of two classes. (Contributed by NM,
14-Dec-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
|
Theorem | ineq2 3276 |
Equality theorem for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
|
Theorem | ineq12 3277 |
Equality theorem for intersection of two classes. (Contributed by NM,
8-May-1994.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
|
Theorem | ineq1i 3278 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) |
|
Theorem | ineq2i 3279 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵) |
|
Theorem | ineq12i 3280 |
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 3281 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
|
Theorem | ineq2d 3282 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
|
Theorem | ineq12d 3283 |
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 3284 |
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 3285 |
A frequently-used variant of subclass definition df-ss 3089. (Contributed
by NM, 10-Jan-2015.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
|
Theorem | dfss5 3286 |
Another definition of subclasshood. Similar to df-ss 3089, dfss 3090, and
dfss1 3285. (Contributed by David Moews, 1-May-2017.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴)) |
|
Theorem | nfin 3287 |
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 3288 |
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 3289* |
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 3290 |
Idempotent law for intersection of classes. Theorem 15 of [Suppes]
p. 26. (Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∩ 𝐴) = 𝐴 |
|
Theorem | inass 3291 |
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 3292 |
A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
|
Theorem | in32 3293 |
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 3294 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴)) |
|
Theorem | in31 3295 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴) |
|
Theorem | inrot 3296 |
Rotate the intersection of 3 classes. (Contributed by NM,
27-Aug-2012.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵) |
|
Theorem | in4 3297 |
Rearrangement of intersection of 4 classes. (Contributed by NM,
21-Apr-2001.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) |
|
Theorem | inindi 3298 |
Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) |
|
Theorem | inindir 3299 |
Intersection distributes over itself. (Contributed by NM,
17-Aug-2004.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) |
|
Theorem | sseqin2 3300 |
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) |