Theorem List for Intuitionistic Logic Explorer - 3001-3100 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | nssne1 3001 |
Two classes are different if they don't include the same class.
(Contributed by NM, 23-Apr-2015.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 ⊆ 𝐶) → 𝐵 ≠ 𝐶) |
|
Theorem | nssne2 3002 |
Two classes are different if they are not subclasses of the same class.
(Contributed by NM, 23-Apr-2015.)
|
⊢ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐵 ⊆ 𝐶) → 𝐴 ≠ 𝐵) |
|
Theorem | nssr 3003* |
Negation of subclass relationship. One direction of Exercise 13 of
[TakeutiZaring] p. 18.
(Contributed by Jim Kingdon, 15-Jul-2018.)
|
⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) → ¬ 𝐴 ⊆ 𝐵) |
|
Theorem | ssralv 3004* |
Quantification restricted to a subclass. (Contributed by NM,
11-Mar-2006.)
|
⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)) |
|
Theorem | ssrexv 3005* |
Existential quantification restricted to a subclass. (Contributed by
NM, 11-Jan-2007.)
|
⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑)) |
|
Theorem | ralss 3006* |
Restricted universal 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 | rexss 3007* |
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 3008 |
Class abstractions in a subclass relationship. (Contributed by NM,
3-Jul-1994.)
|
⊢ ({𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 → 𝜓)) |
|
Theorem | abss 3009* |
Class abstraction in a subclass relationship. (Contributed by NM,
16-Aug-2006.)
|
⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴)) |
|
Theorem | ssab 3010* |
Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
|
⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
|
Theorem | ssabral 3011* |
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 3012 |
Inference of abstraction subclass from implication. (Contributed by NM,
31-Mar-1995.)
|
⊢ (𝜑 → 𝜓) ⇒ ⊢ {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓} |
|
Theorem | ss2abdv 3013* |
Deduction of abstraction subclass from implication. (Contributed by NM,
29-Jul-2011.)
|
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) |
|
Theorem | abssdv 3014* |
Deduction of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
|
⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴) |
|
Theorem | abssi 3015* |
Inference of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
|
⊢ (𝜑 → 𝑥 ∈ 𝐴) ⇒ ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴 |
|
Theorem | ss2rab 3016 |
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 3017* |
Restricted class abstraction in a subclass relationship. (Contributed
by NM, 16-Aug-2006.)
|
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
|
Theorem | ssrab 3018* |
Subclass of a restricted class abstraction. (Contributed by NM,
16-Aug-2006.)
|
⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑)) |
|
Theorem | ssrabdv 3019* |
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 3020* |
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 3021* |
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 3022 |
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 3023* |
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 3024* |
Subclass relation for the restriction of a class abstraction.
(Contributed by NM, 31-Mar-1995.)
|
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
|
Theorem | ssrab2 3025* |
Subclass relation for a restricted class. (Contributed by NM,
19-Mar-1997.)
|
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 |
|
Theorem | ssrabeq 3026* |
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 3027 |
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 3028* |
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 3029 |
Alternate definition of proper subclass. (Contributed by NM,
7-Feb-1996.)
|
⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) |
|
Theorem | dfpss3 3030 |
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 3031 |
Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
|
⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) |
|
Theorem | psseq2 3032 |
Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
|
⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) |
|
Theorem | psseq1i 3033 |
An equality inference for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶) |
|
Theorem | psseq2i 3034 |
An equality inference for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵) |
|
Theorem | psseq12i 3035 |
An equality inference for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ 𝐴 = 𝐵
& ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷) |
|
Theorem | psseq1d 3036 |
An equality deduction for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) |
|
Theorem | psseq2d 3037 |
An equality deduction for the proper subclass relationship.
(Contributed by NM, 9-Jun-2004.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) |
|
Theorem | psseq12d 3038 |
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 3039 |
A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23.
(Contributed by NM, 7-Feb-1996.)
|
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) |
|
Theorem | pssne 3040 |
Two classes in a proper subclass relationship are not equal. (Contributed
by NM, 16-Feb-2015.)
|
⊢ (𝐴 ⊊ 𝐵 → 𝐴 ≠ 𝐵) |
|
Theorem | pssssd 3041 |
Deduce subclass from proper subclass. (Contributed by NM,
29-Feb-1996.)
|
⊢ (𝜑 → 𝐴 ⊊ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
|
Theorem | pssned 3042 |
Proper subclasses are unequal. Deduction form of pssne 3040.
(Contributed by David Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊊ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
|
Theorem | sspssr 3043 |
Subclass in terms of proper subclass. (Contributed by Jim Kingdon,
16-Jul-2018.)
|
⊢ ((𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵) → 𝐴 ⊆ 𝐵) |
|
Theorem | pssirr 3044 |
Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23.
(Contributed by NM, 7-Feb-1996.)
|
⊢ ¬ 𝐴 ⊊ 𝐴 |
|
Theorem | pssn2lp 3045 |
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 3046 |
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 3047 |
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 3048 |
Like pssn2lp 3045 but for subset and proper subset.
(Contributed by Jim
Kingdon, 17-Jul-2018.)
|
⊢ ¬ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐴) |
|
Theorem | psstr 3049 |
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 3050 |
Transitive law for subclass and proper subclass. (Contributed by NM,
3-Apr-1996.)
|
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) |
|
Theorem | psssstr 3051 |
Transitive law for subclass and proper subclass. (Contributed by NM,
3-Apr-1996.)
|
⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) |
|
Theorem | psstrd 3052 |
Proper subclass inclusion is transitive. Deduction form of psstr 3049.
(Contributed by David Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊊ 𝐵)
& ⊢ (𝜑 → 𝐵 ⊊ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
|
Theorem | sspsstrd 3053 |
Transitivity involving subclass and proper subclass inclusion.
Deduction form of sspsstr 3050. (Contributed by David Moews,
1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵)
& ⊢ (𝜑 → 𝐵 ⊊ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
|
Theorem | psssstrd 3054 |
Transitivity involving subclass and proper subclass inclusion.
Deduction form of psssstr 3051. (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 3055 |
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 3056 |
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 3057 |
Equality theorem for class difference. (Contributed by FL,
31-Aug-2009.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) |
|
Theorem | difeq1i 3058 |
Inference adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶) |
|
Theorem | difeq2i 3059 |
Inference adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∖ 𝐴) = (𝐶 ∖ 𝐵) |
|
Theorem | difeq12i 3060 |
Equality inference for class difference. (Contributed by NM,
29-Aug-2004.)
|
⊢ 𝐴 = 𝐵
& ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷) |
|
Theorem | difeq1d 3061 |
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 3062 |
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 3063 |
Equality deduction for class difference. (Contributed by FL,
29-May-2014.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) |
|
Theorem | difeqri 3064* |
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 3065 |
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 3066 |
Implication of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
|
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ 𝐵) |
|
Theorem | eldifn 3067 |
Implication of membership in a class difference. (Contributed by NM,
3-May-1994.)
|
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → ¬ 𝐴 ∈ 𝐶) |
|
Theorem | elndif 3068 |
A set does not belong to a class excluding it. (Contributed by NM,
27-Jun-1994.)
|
⊢ (𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐶 ∖ 𝐵)) |
|
Theorem | difdif 3069 |
Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
(Contributed by NM, 17-May-1998.)
|
⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴 |
|
Theorem | difss 3070 |
Subclass relationship for class difference. Exercise 14 of
[TakeutiZaring] p. 22.
(Contributed by NM, 29-Apr-1994.)
|
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 |
|
Theorem | difssd 3071 |
A difference of two classes is contained in the minuend. Deduction form
of difss 3070. (Contributed by David Moews, 1-May-2017.)
|
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) |
|
Theorem | difss2 3072 |
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 3073 |
If a class is contained in a difference, it is contained in the minuend.
Deduction form of difss2 3072. (Contributed by David Moews,
1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∖ 𝐶)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
|
Theorem | ssdifss 3074 |
Preservation of a subclass relationship by class difference. (Contributed
by NM, 15-Feb-2007.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵) |
|
Theorem | ddifnel 3075* |
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 3175. (Contributed by Jim
Kingdon, 21-Jul-2018.)
|
⊢ (¬ 𝑥 ∈ (V ∖ 𝐴) → 𝑥 ∈ 𝐴) ⇒ ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 |
|
Theorem | ssconb 3076 |
Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
|
⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴))) |
|
Theorem | sscon 3077 |
Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.
(Contributed by NM, 22-Mar-1998.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
|
Theorem | ssdif 3078 |
Difference law for subsets. (Contributed by NM, 28-May-1998.)
|
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
|
Theorem | ssdifd 3079 |
If 𝐴 is contained in 𝐵, then
(𝐴 ∖
𝐶) is contained in
(𝐵
∖ 𝐶).
Deduction form of ssdif 3078. (Contributed by David
Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
|
Theorem | sscond 3080 |
If 𝐴 is contained in 𝐵, then
(𝐶 ∖
𝐵) is contained in
(𝐶
∖ 𝐴).
Deduction form of sscon 3077. (Contributed by David
Moews, 1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∖ 𝐵) ⊆ (𝐶 ∖ 𝐴)) |
|
Theorem | ssdifssd 3081 |
If 𝐴 is contained in 𝐵, then
(𝐴 ∖
𝐶) is also contained
in
𝐵. Deduction form of ssdifss 3074. (Contributed by David Moews,
1-May-2017.)
|
⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ 𝐵) |
|
Theorem | ssdif2d 3082 |
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 3083 |
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 3084 |
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 3085* |
Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
|
⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) ⇒ ⊢ (𝐴 ∪ 𝐵) = 𝐶 |
|
Theorem | unidm 3086 |
Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27.
(Contributed by NM, 5-Aug-1993.)
|
⊢ (𝐴 ∪ 𝐴) = 𝐴 |
|
Theorem | uncom 3087 |
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 3088 |
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 3089 |
Inference form of equncom 3088. (Contributed by Alan Sare,
18-Feb-2012.)
|
⊢ 𝐴 = (𝐵 ∪ 𝐶) ⇒ ⊢ 𝐴 = (𝐶 ∪ 𝐵) |
|
Theorem | uneq1 3090 |
Equality theorem for union of two classes. (Contributed by NM,
5-Aug-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) |
|
Theorem | uneq2 3091 |
Equality theorem for the union of two classes. (Contributed by NM,
5-Aug-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵)) |
|
Theorem | uneq12 3092 |
Equality theorem for union of two classes. (Contributed by NM,
29-Mar-1998.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
|
Theorem | uneq1i 3093 |
Inference adding union to the right in a class equality. (Contributed
by NM, 30-Aug-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶) |
|
Theorem | uneq2i 3094 |
Inference adding union to the left in a class equality. (Contributed by
NM, 30-Aug-1993.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∪ 𝐴) = (𝐶 ∪ 𝐵) |
|
Theorem | uneq12i 3095 |
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 3096 |
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 3097 |
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 3098 |
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 3099 |
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 3100 |
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)) |