Theorem List for Intuitionistic Logic Explorer - 3001-3100 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | rexss 3001* |
Restricted existential quantification on a subset in terms of superset.
(Contributed by Stefan O'Rear, 3-Apr-2015.)
|
⊢ (A ⊆
B → (∃x ∈ A φ ↔ ∃x ∈ B (x ∈ A ∧ φ))) |
|
Theorem | ss2ab 3002 |
Class abstractions in a subclass relationship. (Contributed by NM,
3-Jul-1994.)
|
⊢ ({x ∣
φ} ⊆ {x ∣ ψ} ↔ ∀x(φ → ψ)) |
|
Theorem | abss 3003* |
Class abstraction in a subclass relationship. (Contributed by NM,
16-Aug-2006.)
|
⊢ ({x ∣
φ} ⊆ A ↔ ∀x(φ → x ∈ A)) |
|
Theorem | ssab 3004* |
Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
|
⊢ (A ⊆
{x ∣ φ} ↔ ∀x(x ∈ A → φ)) |
|
Theorem | ssabral 3005* |
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) |
|
Theorem | ss2abi 3006 |
Inference of abstraction subclass from implication. (Contributed by NM,
31-Mar-1995.)
|
⊢ (φ
→ ψ)
⇒ ⊢ {x ∣ φ} ⊆ {x ∣ ψ} |
|
Theorem | ss2abdv 3007* |
Deduction of abstraction subclass from implication. (Contributed by NM,
29-Jul-2011.)
|
⊢ (φ
→ (ψ → χ)) ⇒ ⊢ (φ → {x ∣ ψ} ⊆ {x ∣ χ}) |
|
Theorem | abssdv 3008* |
Deduction of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
|
⊢ (φ
→ (ψ → x ∈ A)) ⇒ ⊢ (φ → {x ∣ ψ} ⊆ A) |
|
Theorem | abssi 3009* |
Inference of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
|
⊢ (φ
→ x ∈ A) ⇒ ⊢ {x ∣ φ} ⊆ A |
|
Theorem | ss2rab 3010 |
Restricted abstraction classes in a subclass relationship. (Contributed
by NM, 30-May-1999.)
|
⊢ ({x ∈ A ∣
φ} ⊆ {x ∈ A ∣ ψ} ↔ ∀x ∈ A (φ → ψ)) |
|
Theorem | rabss 3011* |
Restricted class abstraction in a subclass relationship. (Contributed
by NM, 16-Aug-2006.)
|
⊢ ({x ∈ A ∣
φ} ⊆ B ↔ ∀x ∈ A (φ → x ∈ B)) |
|
Theorem | ssrab 3012* |
Subclass of a restricted class abstraction. (Contributed by NM,
16-Aug-2006.)
|
⊢ (B ⊆
{x ∈
A ∣ φ} ↔ (B ⊆ A
∧ ∀x ∈ B φ)) |
|
Theorem | ssrabdv 3013* |
Subclass of a restricted class abstraction (deduction rule).
(Contributed by NM, 31-Aug-2006.)
|
⊢ (φ
→ B ⊆ A)
& ⊢ ((φ
∧ x ∈ B) →
ψ) ⇒ ⊢ (φ → B ⊆ {x
∈ A
∣ ψ}) |
|
Theorem | rabssdv 3014* |
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) |
|
Theorem | ss2rabdv 3015* |
Deduction of restricted abstraction subclass from implication.
(Contributed by NM, 30-May-2006.)
|
⊢ ((φ
∧ x ∈ A) →
(ψ → χ)) ⇒ ⊢ (φ → {x ∈ A ∣ ψ} ⊆ {x ∈ A ∣ χ}) |
|
Theorem | ss2rabi 3016 |
Inference of restricted abstraction subclass from implication.
(Contributed by NM, 14-Oct-1999.)
|
⊢ (x ∈ A →
(φ → ψ)) ⇒ ⊢ {x ∈ A ∣ φ} ⊆ {x ∈ A ∣ ψ} |
|
Theorem | rabss2 3017* |
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}) |
|
Theorem | ssab2 3018* |
Subclass relation for the restriction of a class abstraction.
(Contributed by NM, 31-Mar-1995.)
|
⊢ {x ∣
(x ∈
A ∧ φ)} ⊆ A |
|
Theorem | ssrab2 3019* |
Subclass relation for a restricted class. (Contributed by NM,
19-Mar-1997.)
|
⊢ {x ∈ A ∣
φ} ⊆ A |
|
Theorem | ssrabeq 3020* |
If the restricting class of a restricted class abstraction is a subset
of this restricted class abstraction, it is equal to this restricted
class abstraction. (Contributed by Alexander van der Vekens,
31-Dec-2017.)
|
⊢ (𝑉 ⊆ {x ∈ 𝑉 ∣ φ} ↔ 𝑉 = {x
∈ 𝑉 ∣ φ}) |
|
Theorem | rabssab 3021 |
A restricted class is a subclass of the corresponding unrestricted class.
(Contributed by Mario Carneiro, 23-Dec-2016.)
|
⊢ {x ∈ A ∣
φ} ⊆ {x ∣ φ} |
|
Theorem | uniiunlem 3022* |
A subset relationship useful for converting union to indexed union using
dfiun2 or dfiun2g and intersection to indexed intersection using
dfiin2 . (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)) |
|
Theorem | dfpss2 3023 |
Alternate definition of proper subclass. (Contributed by NM,
7-Feb-1996.)
|
⊢ (A ⊊
B ↔ (A ⊆ B
∧ ¬ A
= B)) |
|
Theorem | dfpss3 3024 |
Alternate definition of proper subclass. (Contributed by NM,
7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (A ⊊
B ↔ (A ⊆ B
∧ ¬ B
⊆ A)) |
|
Theorem | psseq1 3025 |
Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
|
⊢ (A =
B → (A ⊊ 𝐶 ↔ B ⊊ 𝐶)) |
|
Theorem | psseq2 3026 |
Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
|
⊢ (A =
B → (𝐶 ⊊ A ↔ 𝐶 ⊊ B)) |
|
Theorem | psseq1i 3027 |
An equality inference for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ A =
B ⇒ ⊢ (A ⊊ 𝐶 ↔ B ⊊ 𝐶) |
|
Theorem | psseq2i 3028 |
An equality inference for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ A =
B ⇒ ⊢ (𝐶 ⊊ A ↔ 𝐶 ⊊ B) |
|
Theorem | psseq12i 3029 |
An equality inference for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ A =
B & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (A ⊊ 𝐶 ↔ B ⊊ 𝐷) |
|
Theorem | psseq1d 3030 |
An equality deduction for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ (φ
→ A = B) ⇒ ⊢ (φ → (A ⊊ 𝐶 ↔ B ⊊ 𝐶)) |
|
Theorem | psseq2d 3031 |
An equality deduction for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ (φ
→ A = B) ⇒ ⊢ (φ → (𝐶 ⊊ A ↔ 𝐶 ⊊ B)) |
|
Theorem | psseq12d 3032 |
An equality deduction for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ (φ
→ A = B)
& ⊢ (φ
→ 𝐶 = 𝐷)
⇒ ⊢ (φ → (A ⊊ 𝐶 ↔ B ⊊ 𝐷)) |
|
Theorem | pssss 3033 |
A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23.
(Contributed by NM, 7-Feb-1996.)
|
⊢ (A ⊊
B → A ⊆ B) |
|
Theorem | pssne 3034 |
Two classes in a proper subclass relationship are not equal. (Contributed
by NM, 16-Feb-2015.)
|
⊢ (A ⊊
B → A ≠ B) |
|
Theorem | pssssd 3035 |
Deduce subclass from proper subclass. (Contributed by NM,
29-Feb-1996.)
|
⊢ (φ
→ A ⊊ B) ⇒ ⊢ (φ → A ⊆ B) |
|
Theorem | pssned 3036 |
Proper subclasses are unequal. Deduction form of pssne 3034.
(Contributed by David Moews, 1-May-2017.)
|
⊢ (φ
→ A ⊊ B) ⇒ ⊢ (φ → A ≠ B) |
|
Theorem | sspssr 3037 |
Subclass in terms of proper subclass. (Contributed by Jim Kingdon,
16-Jul-2018.)
|
⊢ ((A
⊊ B
∨ A = B) → A
⊆ B) |
|
Theorem | pssirr 3038 |
Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23.
(Contributed by NM, 7-Feb-1996.)
|
⊢ ¬ A
⊊ A |
|
Theorem | pssn2lp 3039 |
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.)
|
⊢ ¬ (A
⊊ B ∧ B ⊊
A) |
|
Theorem | sspsstrir 3040 |
Two ways of stating trichotomy with respect to inclusion. (Contributed by
Jim Kingdon, 16-Jul-2018.)
|
⊢ ((A
⊊ B
∨ A = B ∨ B ⊊ A)
→ (A ⊆ B ∨ B ⊆ A)) |
|
Theorem | ssnpss 3041 |
Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.)
(Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (A ⊆
B → ¬ B ⊊ A) |
|
Theorem | sspssn 3042 |
Like pssn2lp 3039 but for subset and proper subset.
(Contributed by Jim
Kingdon, 17-Jul-2018.)
|
⊢ ¬ (A
⊆ B ∧ B ⊊
A) |
|
Theorem | psstr 3043 |
Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23.
(Contributed by NM, 7-Feb-1996.)
|
⊢ ((A
⊊ B ∧ B ⊊
𝐶) → A ⊊ 𝐶) |
|
Theorem | sspsstr 3044 |
Transitive law for subclass and proper subclass. (Contributed by NM,
3-Apr-1996.)
|
⊢ ((A ⊆
B ∧
B ⊊ 𝐶) → A ⊊ 𝐶) |
|
Theorem | psssstr 3045 |
Transitive law for subclass and proper subclass. (Contributed by NM,
3-Apr-1996.)
|
⊢ ((A
⊊ B ∧ B ⊆
𝐶) → A ⊊ 𝐶) |
|
Theorem | psstrd 3046 |
Proper subclass inclusion is transitive. Deduction form of psstr 3043.
(Contributed by David Moews, 1-May-2017.)
|
⊢ (φ
→ A ⊊ B)
& ⊢ (φ
→ B ⊊ 𝐶) ⇒ ⊢ (φ → A ⊊ 𝐶) |
|
Theorem | sspsstrd 3047 |
Transitivity involving subclass and proper subclass inclusion.
Deduction form of sspsstr 3044. (Contributed by David Moews,
1-May-2017.)
|
⊢ (φ
→ A ⊆ B)
& ⊢ (φ
→ B ⊊ 𝐶) ⇒ ⊢ (φ → A ⊊ 𝐶) |
|
Theorem | psssstrd 3048 |
Transitivity involving subclass and proper subclass inclusion.
Deduction form of psssstr 3045. (Contributed by David Moews,
1-May-2017.)
|
⊢ (φ
→ A ⊊ B)
& ⊢ (φ
→ B ⊆ 𝐶) ⇒ ⊢ (φ → A ⊊ 𝐶) |
|
2.1.13 The difference, union, and intersection
of two classes
|
|
2.1.13.1 The difference of two
classes
|
|
Theorem | difeq1 3049 |
Equality theorem for class difference. (Contributed by NM,
10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (A =
B → (A ∖ 𝐶) = (B
∖ 𝐶)) |
|
Theorem | difeq2 3050 |
Equality theorem for class difference. (Contributed by NM,
10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|
⊢ (A =
B → (𝐶 ∖ A) = (𝐶 ∖ B)) |
|
Theorem | difeq12 3051 |
Equality theorem for class difference. (Contributed by FL,
31-Aug-2009.)
|
⊢ ((A =
B ∧ 𝐶 = 𝐷) → (A ∖ 𝐶) = (B
∖ 𝐷)) |
|
Theorem | difeq1i 3052 |
Inference adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
|
⊢ A =
B ⇒ ⊢ (A ∖ 𝐶) = (B
∖ 𝐶) |
|
Theorem | difeq2i 3053 |
Inference adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
|
⊢ A =
B ⇒ ⊢ (𝐶 ∖ A) = (𝐶 ∖ B) |
|
Theorem | difeq12i 3054 |
Equality inference for class difference. (Contributed by NM,
29-Aug-2004.)
|
⊢ A =
B & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (A ∖ 𝐶) = (B
∖ 𝐷) |
|
Theorem | difeq1d 3055 |
Deduction adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
|
⊢ (φ
→ A = B) ⇒ ⊢ (φ → (A ∖ 𝐶) = (B
∖ 𝐶)) |
|
Theorem | difeq2d 3056 |
Deduction adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
|
⊢ (φ
→ A = B) ⇒ ⊢ (φ → (𝐶 ∖ A) = (𝐶 ∖ B)) |
|
Theorem | difeq12d 3057 |
Equality deduction for class difference. (Contributed by FL,
29-May-2014.)
|
⊢ (φ
→ A = B)
& ⊢ (φ
→ 𝐶 = 𝐷)
⇒ ⊢ (φ → (A ∖ 𝐶) = (B
∖ 𝐷)) |
|
Theorem | difeqri 3058* |
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 3059 |
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 3060 |
Implication of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
|
⊢ (A ∈ (B ∖
𝐶) → A ∈ B) |
|
Theorem | eldifn 3061 |
Implication of membership in a class difference. (Contributed by NM,
3-May-1994.)
|
⊢ (A ∈ (B ∖
𝐶) → ¬ A ∈ 𝐶) |
|
Theorem | elndif 3062 |
A set does not belong to a class excluding it. (Contributed by NM,
27-Jun-1994.)
|
⊢ (A ∈ B →
¬ A ∈ (𝐶 ∖ B)) |
|
Theorem | difdif 3063 |
Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
(Contributed by NM, 17-May-1998.)
|
⊢ (A ∖
(B ∖ A)) = A |
|
Theorem | difss 3064 |
Subclass relationship for class difference. Exercise 14 of
[TakeutiZaring] p. 22.
(Contributed by NM, 29-Apr-1994.)
|
⊢ (A ∖
B) ⊆ A |
|
Theorem | difssd 3065 |
A difference of two classes is contained in the minuend. Deduction form
of difss 3064. (Contributed by David Moews, 1-May-2017.)
|
⊢ (φ
→ (A ∖ B) ⊆ A) |
|
Theorem | difss2 3066 |
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 3067 |
If a class is contained in a difference, it is contained in the
minuend. Deduction form of difss2 3066. (Contributed by David Moews,
1-May-2017.)
|
⊢ (φ
→ A ⊆ (B ∖ 𝐶)) ⇒ ⊢ (φ → A ⊆ B) |
|
Theorem | ssdifss 3068 |
Preservation of a subclass relationship by class difference. (Contributed
by NM, 15-Feb-2007.)
|
⊢ (A ⊆
B → (A ∖ 𝐶) ⊆ B) |
|
Theorem | ddifnel 3069* |
Double complement under universal class. The hypothesis is one way of
expressing the idea that membership in A is decidable. 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 3169. (Contributed by Jim
Kingdon, 21-Jul-2018.)
|
⊢ (¬ x
∈ (V ∖ A) → x
∈ A) ⇒ ⊢ (V ∖ (V ∖ A)) = A |
|
Theorem | ssconb 3070 |
Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
|
⊢ ((A ⊆
𝐶 ∧ B ⊆
𝐶) → (A ⊆ (𝐶 ∖ B) ↔ B
⊆ (𝐶 ∖
A))) |
|
Theorem | sscon 3071 |
Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
(Contributed by NM, 22-Mar-1998.)
|
⊢ (A ⊆
B → (𝐶 ∖ B) ⊆ (𝐶 ∖ A)) |
|
Theorem | ssdif 3072 |
Difference law for subsets. (Contributed by NM, 28-May-1998.)
|
⊢ (A ⊆
B → (A ∖ 𝐶) ⊆ (B ∖ 𝐶)) |
|
Theorem | ssdifd 3073 |
If A is contained
in B, then (A ∖ 𝐶) is contained in
(B ∖ 𝐶). Deduction form of ssdif 3072. (Contributed by David
Moews, 1-May-2017.)
|
⊢ (φ
→ A ⊆ B) ⇒ ⊢ (φ → (A ∖ 𝐶) ⊆ (B ∖ 𝐶)) |
|
Theorem | sscond 3074 |
If A is contained
in B, then (𝐶 ∖
B) is contained in
(𝐶
∖ A). Deduction form of sscon 3071. (Contributed by David
Moews, 1-May-2017.)
|
⊢ (φ
→ A ⊆ B) ⇒ ⊢ (φ → (𝐶 ∖ B) ⊆ (𝐶 ∖ A)) |
|
Theorem | ssdifssd 3075 |
If A is contained
in B, then (A ∖ 𝐶) is also contained in
B.
Deduction form of ssdifss 3068. (Contributed by David Moews,
1-May-2017.)
|
⊢ (φ
→ A ⊆ B) ⇒ ⊢ (φ → (A ∖ 𝐶) ⊆ B) |
|
Theorem | ssdif2d 3076 |
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 3077 |
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 3078 |
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 3079* |
Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
|
⊢ ((x ∈ A ∨ x ∈ B) ↔
x ∈
𝐶) ⇒ ⊢ (A ∪ B) =
𝐶 |
|
Theorem | unidm 3080 |
Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
(Contributed by NM, 5-Aug-1993.)
|
⊢ (A ∪
A) = A |
|
Theorem | uncom 3081 |
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 3082 |
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 3083 |
Inference form of equncom 3082. (Contributed by Alan Sare,
18-Feb-2012.)
|
⊢ A =
(B ∪ 𝐶) ⇒ ⊢ A = (𝐶 ∪ B) |
|
Theorem | uneq1 3084 |
Equality theorem for union of two classes. (Contributed by NM,
5-Aug-1993.)
|
⊢ (A =
B → (A ∪ 𝐶) = (B
∪ 𝐶)) |
|
Theorem | uneq2 3085 |
Equality theorem for the union of two classes. (Contributed by NM,
5-Aug-1993.)
|
⊢ (A =
B → (𝐶 ∪ A) = (𝐶 ∪ B)) |
|
Theorem | uneq12 3086 |
Equality theorem for union of two classes. (Contributed by NM,
29-Mar-1998.)
|
⊢ ((A =
B ∧ 𝐶 = 𝐷) → (A ∪ 𝐶) = (B
∪ 𝐷)) |
|
Theorem | uneq1i 3087 |
Inference adding union to the right in a class equality. (Contributed
by NM, 30-Aug-1993.)
|
⊢ A =
B ⇒ ⊢ (A ∪ 𝐶) = (B
∪ 𝐶) |
|
Theorem | uneq2i 3088 |
Inference adding union to the left in a class equality. (Contributed by
NM, 30-Aug-1993.)
|
⊢ A =
B ⇒ ⊢ (𝐶 ∪ A) = (𝐶 ∪ B) |
|
Theorem | uneq12i 3089 |
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 3090 |
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 3091 |
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 3092 |
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 3093 |
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 3094 |
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 3095 |
A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
|
⊢ (A ∪
(B ∪ 𝐶)) = (B
∪ (A ∪ 𝐶)) |
|
Theorem | un23 3096 |
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 3097 |
A rearrangement of the union of 4 classes. (Contributed by NM,
12-Aug-2004.)
|
⊢ ((A ∪
B) ∪ (𝐶 ∪ 𝐷)) = ((A ∪ 𝐶) ∪ (B ∪ 𝐷)) |
|
Theorem | unundi 3098 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|
⊢ (A ∪
(B ∪ 𝐶)) = ((A ∪ B)
∪ (A ∪ 𝐶)) |
|
Theorem | unundir 3099 |
Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
|
⊢ ((A ∪
B) ∪ 𝐶) = ((A
∪ 𝐶) ∪ (B ∪ 𝐶)) |
|
Theorem | ssun1 3100 |
Subclass relationship for union of classes. Theorem 25 of [Suppes]
p. 27. (Contributed by NM, 5-Aug-1993.)
|
⊢ A ⊆
(A ∪ B) |