Theorem List for Intuitionistic Logic Explorer - 3101-3200 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | difss2d 3101 |
If a class is contained in a difference, it is contained in the minuend.
Deduction form of difss2 3100. (Contributed by David Moews,
1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
|
Theorem | ssdifss 3102 |
Preservation of a subclass relationship by class difference. (Contributed
by NM, 15-Feb-2007.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵) |
|
Theorem | ddifnel 3103* |
Double complement under universal class. The hypothesis corresponds to
stability of membership in 𝐴, which is weaker than decidability
(see dcimpstab 785). Actually, the conclusion is a
characterization of
stability of membership in a class (see ddifstab 3104) . 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 3202. (Contributed by Jim Kingdon,
21-Jul-2018.)
|
⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) → 𝑥 ∈ 𝐴) ⇒ ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 |
|
Theorem | ddifstab 3104* |
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 3105 |
Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
|
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴))) |
|
Theorem | sscon 3106 |
Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
(Contributed by NM, 22-Mar-1998.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
|
Theorem | ssdif 3107 |
Difference law for subsets. (Contributed by NM, 28-May-1998.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
|
Theorem | ssdifd 3108 |
If 𝐴 is contained in 𝐵, then
(𝐴 ∖
𝐶) is contained in
(𝐵
∖ 𝐶).
Deduction form of ssdif 3107. (Contributed by David
Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
|
Theorem | sscond 3109 |
If 𝐴 is contained in 𝐵, then
(𝐶 ∖
𝐵) is contained in
(𝐶
∖ 𝐴).
Deduction form of sscon 3106. (Contributed by David
Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
|
Theorem | ssdifssd 3110 |
If 𝐴 is contained in 𝐵, then
(𝐴 ∖
𝐶) is also contained
in
𝐵. Deduction form of ssdifss 3102. (Contributed by David Moews,
1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵) |
|
Theorem | ssdif2d 3111 |
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 3112 |
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 3113 |
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 3114* |
Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
|
⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∪ 𝐵) = 𝐶 |
|
Theorem | unidm 3115 |
Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
(Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∪ 𝐴) = 𝐴 |
|
Theorem | uncom 3116 |
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 3117 |
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 3118 |
Inference form of equncom 3117. (Contributed by Alan Sare,
18-Feb-2012.)
|
⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ 𝐴 = (𝐶 ∪ 𝐵) |
|
Theorem | uneq1 3119 |
Equality theorem for union of two classes. (Contributed by NM,
5-Aug-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
|
Theorem | uneq2 3120 |
Equality theorem for the union of two classes. (Contributed by NM,
5-Aug-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) |
|
Theorem | uneq12 3121 |
Equality theorem for union of two classes. (Contributed by NM,
29-Mar-1998.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
|
Theorem | uneq1i 3122 |
Inference adding union to the right in a class equality. (Contributed
by NM, 30-Aug-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) |
|
Theorem | uneq2i 3123 |
Inference adding union to the left in a class equality. (Contributed by
NM, 30-Aug-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵) |
|
Theorem | uneq12i 3124 |
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 3125 |
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 3126 |
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 3127 |
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 3128 |
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 3129 |
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 3130 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
|
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) |
|
Theorem | un23 3131 |
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 3132 |
A rearrangement of the union of 4 classes. (Contributed by NM,
12-Aug-2004.)
|
⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐷)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐷)) |
|
Theorem | unundi 3133 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|
⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶)) |
|
Theorem | unundir 3134 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|
⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶)) |
|
Theorem | ssun1 3135 |
Subclass relationship for union of classes. Theorem 25 of [Suppes]
p. 27. (Contributed by NM, 5-Aug-1993.)
|
⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵) |
|
Theorem | ssun2 3136 |
Subclass relationship for union of classes. (Contributed by NM,
30-Aug-1993.)
|
⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) |
|
Theorem | ssun3 3137 |
Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶)) |
|
Theorem | ssun4 3138 |
Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
|
⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
|
Theorem | elun1 3139 |
Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐵 ∪ 𝐶)) |
|
Theorem | elun2 3140 |
Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
|
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (𝐶 ∪ 𝐵)) |
|
Theorem | unss1 3141 |
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 3142 |
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 3143 |
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 3144 |
Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷)) |
|
Theorem | ssequn2 3145 |
A relationship between subclass and union. (Contributed by NM,
13-Jun-1994.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐴) = 𝐵) |
|
Theorem | unss 3146 |
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 3147 |
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 3148 |
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 3149 |
If (𝐴
∪ 𝐵) is
contained in 𝐶, so is 𝐴. One-way
deduction form of unss 3146. Partial converse of unssd 3148. (Contributed
by David Moews, 1-May-2017.)
|
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
|
Theorem | unssbd 3150 |
If (𝐴
∪ 𝐵) is
contained in 𝐶, so is 𝐵. One-way
deduction form of unss 3146. Partial converse of unssd 3148. (Contributed
by David Moews, 1-May-2017.)
|
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
|
Theorem | ssun 3151 |
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 3152 |
Restricted existential quantification over union. (Contributed by Jeff
Madsen, 5-Jan-2011.)
|
⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑)) |
|
Theorem | ralunb 3153 |
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 3154 |
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 3155 |
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 | elin2 3156 |
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 | elin3 3157 |
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 3158 |
Commutative law for intersection of classes. Exercise 7 of
[TakeutiZaring] p. 17.
(Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) |
|
Theorem | ineqri 3159* |
Inference from membership to intersection. (Contributed by NM,
5-Aug-1993.)
|
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
|
Theorem | ineq1 3160 |
Equality theorem for intersection of two classes. (Contributed by NM,
14-Dec-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
|
Theorem | ineq2 3161 |
Equality theorem for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
|
Theorem | ineq12 3162 |
Equality theorem for intersection of two classes. (Contributed by NM,
8-May-1994.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
|
Theorem | ineq1i 3163 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) |
|
Theorem | ineq2i 3164 |
Equality inference for intersection of two classes. (Contributed by NM,
26-Dec-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵) |
|
Theorem | ineq12i 3165 |
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 3166 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
|
Theorem | ineq2d 3167 |
Equality deduction for intersection of two classes. (Contributed by NM,
10-Apr-1994.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
|
Theorem | ineq12d 3168 |
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 3169 |
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 3170 |
A frequently-used variant of subclass definition df-ss 2986. (Contributed
by NM, 10-Jan-2015.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
|
Theorem | dfss5 3171 |
Another definition of subclasshood. Similar to df-ss 2986, dfss 2987, and
dfss1 3170. (Contributed by David Moews, 1-May-2017.)
|
⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴)) |
|
Theorem | nfin 3172 |
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 3173 |
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 3174* |
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 3175 |
Idempotent law for intersection of classes. Theorem 15 of [Suppes]
p. 26. (Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∩ 𝐴) = 𝐴 |
|
Theorem | inass 3176 |
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 3177 |
A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
|
Theorem | in32 3178 |
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 3179 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴)) |
|
Theorem | in31 3180 |
A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴) |
|
Theorem | inrot 3181 |
Rotate the intersection of 3 classes. (Contributed by NM,
27-Aug-2012.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵) |
|
Theorem | in4 3182 |
Rearrangement of intersection of 4 classes. (Contributed by NM,
21-Apr-2001.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) |
|
Theorem | inindi 3183 |
Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
|
⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) |
|
Theorem | inindir 3184 |
Intersection distributes over itself. (Contributed by NM,
17-Aug-2004.)
|
⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) |
|
Theorem | sseqin2 3185 |
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 3186 |
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 3187 |
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 3188 |
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 3189 |
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 3190 |
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 3191 |
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 3192 |
Add left intersection to subclass relation. (Contributed by NM,
19-Oct-1999.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵)) |
|
Theorem | ss2in 3193 |
Intersection of subclasses. (Contributed by NM, 5-May-2000.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷)) |
|
Theorem | ssinss1 3194 |
Intersection preserves subclass relationship. (Contributed by NM,
14-Sep-1999.)
|
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
|
Theorem | inss 3195 |
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 3196 |
Absorption law for union. (Contributed by NM, 16-Apr-2006.)
|
⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴 |
|
Theorem | inabs 3197 |
Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
|
⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴 |
|
Theorem | ssddif 3198 |
Double complement and subset. Similar to ddifss 3202 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 3199 |
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 3200 |
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)) |