Theorem List for Intuitionistic Logic Explorer - 3901-4000 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | nfiu1 3901 |
Bound-variable hypothesis builder for indexed union. (Contributed by
NM, 12-Oct-2003.)
|
⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 |
|
Theorem | nfii1 3902 |
Bound-variable hypothesis builder for indexed intersection.
(Contributed by NM, 15-Oct-2003.)
|
⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 |
|
Theorem | dfiun2g 3903* |
Alternate definition of indexed union when 𝐵 is a set. Definition
15(a) of [Suppes] p. 44. (Contributed by
NM, 23-Mar-2006.) (Proof
shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪
𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
|
Theorem | dfiin2g 3904* |
Alternate definition of indexed intersection when 𝐵 is a set.
(Contributed by Jeff Hankins, 27-Aug-2009.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩
𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
|
Theorem | dfiun2 3905* |
Alternate definition of indexed union when 𝐵 is a set. Definition
15(a) of [Suppes] p. 44. (Contributed by
NM, 27-Jun-1998.) (Revised by
David Abernethy, 19-Jun-2012.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
|
Theorem | dfiin2 3906* |
Alternate definition of indexed intersection when 𝐵 is a set.
Definition 15(b) of [Suppes] p. 44.
(Contributed by NM, 28-Jun-1998.)
(Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
|
Theorem | dfiunv2 3907* |
Define double indexed union. (Contributed by FL, 6-Nov-2013.)
|
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶} |
|
Theorem | cbviun 3908* |
Rule used to change the bound variables in an indexed union, with the
substitution specified implicitly by the hypothesis. (Contributed by
NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)
|
⊢ Ⅎ𝑦𝐵
& ⊢ Ⅎ𝑥𝐶
& ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪
𝑦 ∈ 𝐴 𝐶 |
|
Theorem | cbviin 3909* |
Change bound variables in an indexed intersection. (Contributed by Jeff
Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
|
⊢ Ⅎ𝑦𝐵
& ⊢ Ⅎ𝑥𝐶
& ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩
𝑦 ∈ 𝐴 𝐶 |
|
Theorem | cbviunv 3910* |
Rule used to change the bound variables in an indexed union, with the
substitution specified implicitly by the hypothesis. (Contributed by
NM, 15-Sep-2003.)
|
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪
𝑦 ∈ 𝐴 𝐶 |
|
Theorem | cbviinv 3911* |
Change bound variables in an indexed intersection. (Contributed by Jeff
Hankins, 26-Aug-2009.)
|
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩
𝑦 ∈ 𝐴 𝐶 |
|
Theorem | iunss 3912* |
Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.)
(Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
|
Theorem | ssiun 3913* |
Subset implication for an indexed union. (Contributed by NM,
3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ (∃𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
|
Theorem | ssiun2 3914 |
Identity law for subset of an indexed union. (Contributed by NM,
12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
|
Theorem | ssiun2s 3915* |
Subset relationship for an indexed union. (Contributed by NM,
26-Oct-2003.)
|
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) ⇒ ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
|
Theorem | iunss2 3916* |
A subclass condition on the members of two indexed classes 𝐶(𝑥)
and 𝐷(𝑦) that implies a subclass relation on
their indexed
unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59.
Compare uniss2 3825. (Contributed by NM, 9-Dec-2004.)
|
⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪
𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷) |
|
Theorem | iunab 3917* |
The indexed union of a class abstraction. (Contributed by NM,
27-Dec-2004.)
|
⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
|
Theorem | iunrab 3918* |
The indexed union of a restricted class abstraction. (Contributed by
NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
|
⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐴 𝜑} |
|
Theorem | iunxdif2 3919* |
Indexed union with a class difference as its index. (Contributed by NM,
10-Dec-2004.)
|
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → ∪
𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 = ∪
𝑥 ∈ 𝐴 𝐶) |
|
Theorem | ssiinf 3920 |
Subset theorem for an indexed intersection. (Contributed by FL,
15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
|
⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) |
|
Theorem | ssiin 3921* |
Subset theorem for an indexed intersection. (Contributed by NM,
15-Oct-2003.)
|
⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) |
|
Theorem | iinss 3922* |
Subset implication for an indexed intersection. (Contributed by NM,
15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩
𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
|
Theorem | iinss2 3923 |
An indexed intersection is included in any of its members. (Contributed
by FL, 15-Oct-2012.)
|
⊢ (𝑥 ∈ 𝐴 → ∩
𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵) |
|
Theorem | uniiun 3924* |
Class union in terms of indexed union. Definition in [Stoll] p. 43.
(Contributed by NM, 28-Jun-1998.)
|
⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 |
|
Theorem | intiin 3925* |
Class intersection in terms of indexed intersection. Definition in
[Stoll] p. 44. (Contributed by NM,
28-Jun-1998.)
|
⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 |
|
Theorem | iunid 3926* |
An indexed union of singletons recovers the index set. (Contributed by
NM, 6-Sep-2005.)
|
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 |
|
Theorem | iun0 3927 |
An indexed union of the empty set is empty. (Contributed by NM,
26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ ∪ 𝑥 ∈ 𝐴 ∅ = ∅ |
|
Theorem | 0iun 3928 |
An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.)
(Proof shortened by Andrew Salmon, 25-Jul-2011.)
|
⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅ |
|
Theorem | 0iin 3929 |
An empty indexed intersection is the universal class. (Contributed by
NM, 20-Oct-2005.)
|
⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V |
|
Theorem | viin 3930* |
Indexed intersection with a universal index class. (Contributed by NM,
11-Sep-2008.)
|
⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴} |
|
Theorem | iunn0m 3931* |
There is an inhabited class in an indexed collection 𝐵(𝑥) iff the
indexed union of them is inhabited. (Contributed by Jim Kingdon,
16-Aug-2018.)
|
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ ∪
𝑥 ∈ 𝐴 𝐵) |
|
Theorem | iinab 3932* |
Indexed intersection of a class builder. (Contributed by NM,
6-Dec-2011.)
|
⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} |
|
Theorem | iinrabm 3933* |
Indexed intersection of a restricted class builder. (Contributed by Jim
Kingdon, 16-Aug-2018.)
|
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩
𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑}) |
|
Theorem | iunin2 3934* |
Indexed union of intersection. Generalization of half of theorem
"Distributive laws" in [Enderton] p. 30. Use uniiun 3924 to recover
Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
|
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪
𝑥 ∈ 𝐴 𝐶) |
|
Theorem | iunin1 3935* |
Indexed union of intersection. Generalization of half of theorem
"Distributive laws" in [Enderton] p. 30. Use uniiun 3924 to recover
Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
|
⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪
𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) |
|
Theorem | iundif2ss 3936* |
Indexed union of class difference. Compare to theorem "De Morgan's
laws" in [Enderton] p. 31.
(Contributed by Jim Kingdon,
17-Aug-2018.)
|
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ⊆ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶) |
|
Theorem | 2iunin 3937* |
Rearrange indexed unions over intersection. (Contributed by NM,
18-Dec-2008.)
|
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (∪
𝑥 ∈ 𝐴 𝐶 ∩ ∪
𝑦 ∈ 𝐵 𝐷) |
|
Theorem | iindif2m 3938* |
Indexed intersection of class difference. Compare to Theorem "De
Morgan's laws" in [Enderton] p.
31. (Contributed by Jim Kingdon,
17-Aug-2018.)
|
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩
𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶)) |
|
Theorem | iinin2m 3939* |
Indexed intersection of intersection. Compare to Theorem "Distributive
laws" in [Enderton] p. 30.
(Contributed by Jim Kingdon,
17-Aug-2018.)
|
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩
𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩
𝑥 ∈ 𝐴 𝐶)) |
|
Theorem | iinin1m 3940* |
Indexed intersection of intersection. Compare to Theorem "Distributive
laws" in [Enderton] p. 30.
(Contributed by Jim Kingdon,
17-Aug-2018.)
|
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩
𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∩
𝑥 ∈ 𝐴 𝐶 ∩ 𝐵)) |
|
Theorem | elriin 3941* |
Elementhood in a relative intersection. (Contributed by Mario Carneiro,
30-Dec-2016.)
|
⊢ (𝐵 ∈ (𝐴 ∩ ∩
𝑥 ∈ 𝑋 𝑆) ↔ (𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑆)) |
|
Theorem | riin0 3942* |
Relative intersection of an empty family. (Contributed by Stefan
O'Rear, 3-Apr-2015.)
|
⊢ (𝑋 = ∅ → (𝐴 ∩ ∩
𝑥 ∈ 𝑋 𝑆) = 𝐴) |
|
Theorem | riinm 3943* |
Relative intersection of an inhabited family. (Contributed by Jim
Kingdon, 19-Aug-2018.)
|
⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩
𝑥 ∈ 𝑋 𝑆) = ∩
𝑥 ∈ 𝑋 𝑆) |
|
Theorem | iinxsng 3944* |
A singleton index picks out an instance of an indexed intersection's
argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario
Carneiro, 17-Nov-2016.)
|
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ∩
𝑥 ∈ {𝐴}𝐵 = 𝐶) |
|
Theorem | iinxprg 3945* |
Indexed intersection with an unordered pair index. (Contributed by NM,
25-Jan-2012.)
|
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
& ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∩ 𝐸)) |
|
Theorem | iunxsng 3946* |
A singleton index picks out an instance of an indexed union's argument.
(Contributed by Mario Carneiro, 25-Jun-2016.)
|
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ∪
𝑥 ∈ {𝐴}𝐵 = 𝐶) |
|
Theorem | iunxsn 3947* |
A singleton index picks out an instance of an indexed union's argument.
(Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro,
25-Jun-2016.)
|
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
|
Theorem | iunxsngf 3948* |
A singleton index picks out an instance of an indexed union's argument.
(Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry
Arnoux, 2-May-2020.)
|
⊢ Ⅎ𝑥𝐶
& ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝐴 ∈ 𝑉 → ∪
𝑥 ∈ {𝐴}𝐵 = 𝐶) |
|
Theorem | iunun 3949 |
Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.)
(Proof shortened by Mario Carneiro, 17-Nov-2016.)
|
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (∪
𝑥 ∈ 𝐴 𝐵 ∪ ∪
𝑥 ∈ 𝐴 𝐶) |
|
Theorem | iunxun 3950 |
Separate a union in the index of an indexed union. (Contributed by NM,
26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
|
⊢ ∪ 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 = (∪
𝑥 ∈ 𝐴 𝐶 ∪ ∪
𝑥 ∈ 𝐵 𝐶) |
|
Theorem | iunxprg 3951* |
A pair index picks out two instances of an indexed union's argument.
(Contributed by Alexander van der Vekens, 2-Feb-2018.)
|
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
& ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷 ∪ 𝐸)) |
|
Theorem | iunxiun 3952* |
Separate an indexed union in the index of an indexed union.
(Contributed by Mario Carneiro, 5-Dec-2016.)
|
⊢ ∪ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 = ∪
𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 |
|
Theorem | iinuniss 3953* |
A relationship involving union and indexed intersection. Exercise 23 of
[Enderton] p. 33 but with equality
changed to subset. (Contributed by
Jim Kingdon, 19-Aug-2018.)
|
⊢ (𝐴 ∪ ∩ 𝐵) ⊆ ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) |
|
Theorem | iununir 3954* |
A relationship involving union and indexed union. Exercise 25 of
[Enderton] p. 33 but with biconditional
changed to implication.
(Contributed by Jim Kingdon, 19-Aug-2018.)
|
⊢ ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) → (𝐵 = ∅ → 𝐴 = ∅)) |
|
Theorem | sspwuni 3955 |
Subclass relationship for power class and union. (Contributed by NM,
18-Jul-2006.)
|
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) |
|
Theorem | pwssb 3956* |
Two ways to express a collection of subclasses. (Contributed by NM,
19-Jul-2006.)
|
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
|
Theorem | elpwpw 3957 |
Characterization of the elements of a double power class: they are exactly
the sets whose union is included in that class. (Contributed by BJ,
29-Apr-2021.)
|
⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴
⊆ 𝐵)) |
|
Theorem | pwpwab 3958* |
The double power class written as a class abstraction: the class of sets
whose union is included in the given class. (Contributed by BJ,
29-Apr-2021.)
|
⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴} |
|
Theorem | pwpwssunieq 3959* |
The class of sets whose union is equal to a given class is included in
the double power class of that class. (Contributed by BJ,
29-Apr-2021.)
|
⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴 |
|
Theorem | elpwuni 3960 |
Relationship for power class and union. (Contributed by NM,
18-Jul-2006.)
|
⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
|
Theorem | iinpw 3961* |
The power class of an intersection in terms of indexed intersection.
Exercise 24(a) of [Enderton] p. 33.
(Contributed by NM,
29-Nov-2003.)
|
⊢ 𝒫 ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 |
|
Theorem | iunpwss 3962* |
Inclusion of an indexed union of a power class in the power class of the
union of its index. Part of Exercise 24(b) of [Enderton] p. 33.
(Contributed by NM, 25-Nov-2003.)
|
⊢ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ⊆ 𝒫 ∪ 𝐴 |
|
Theorem | rintm 3963* |
Relative intersection of an inhabited class. (Contributed by Jim
Kingdon, 19-Aug-2018.)
|
⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) |
|
2.1.21 Disjointness
|
|
Syntax | wdisj 3964 |
Extend wff notation to include the statement that a family of classes
𝐵(𝑥), for 𝑥 ∈ 𝐴, is a disjoint family.
|
wff Disj 𝑥 ∈ 𝐴 𝐵 |
|
Definition | df-disj 3965* |
A collection of classes 𝐵(𝑥) is disjoint when for each element
𝑦, it is in 𝐵(𝑥) for at most one 𝑥.
(Contributed by
Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
|
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
|
Theorem | dfdisj2 3966* |
Alternate definition for disjoint classes. (Contributed by NM,
17-Jun-2017.)
|
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
|
Theorem | disjss2 3967 |
If each element of a collection is contained in a disjoint collection,
the original collection is also disjoint. (Contributed by Mario
Carneiro, 14-Nov-2016.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵)) |
|
Theorem | disjeq2 3968 |
Equality theorem for disjoint collection. (Contributed by Mario Carneiro,
14-Nov-2016.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶)) |
|
Theorem | disjeq2dv 3969* |
Equality deduction for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶)) |
|
Theorem | disjss1 3970* |
A subset of a disjoint collection is disjoint. (Contributed by Mario
Carneiro, 14-Nov-2016.)
|
⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶)) |
|
Theorem | disjeq1 3971* |
Equality theorem for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
|
⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) |
|
Theorem | disjeq1d 3972* |
Equality theorem for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) |
|
Theorem | disjeq12d 3973* |
Equality theorem for disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
|
Theorem | cbvdisj 3974* |
Change bound variables in a disjoint collection. (Contributed by Mario
Carneiro, 14-Nov-2016.)
|
⊢ Ⅎ𝑦𝐵
& ⊢ Ⅎ𝑥𝐶
& ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) |
|
Theorem | cbvdisjv 3975* |
Change bound variables in a disjoint collection. (Contributed by Mario
Carneiro, 11-Dec-2016.)
|
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) |
|
Theorem | nfdisjv 3976* |
Bound-variable hypothesis builder for disjoint collection. (Contributed
by Jim Kingdon, 19-Aug-2018.)
|
⊢ Ⅎ𝑦𝐴
& ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 |
|
Theorem | nfdisj1 3977 |
Bound-variable hypothesis builder for disjoint collection. (Contributed
by Mario Carneiro, 14-Nov-2016.)
|
⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵 |
|
Theorem | disjnim 3978* |
If a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint, then pairs are
disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by
Jim Kingdon, 6-Oct-2022.)
|
⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶) ⇒ ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (𝐵 ∩ 𝐶) = ∅)) |
|
Theorem | disjnims 3979* |
If a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint, then pairs are
disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by
Jim Kingdon, 7-Oct-2022.)
|
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) |
|
Theorem | disji2 3980* |
Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and
𝐵(𝑌) = 𝐷, and 𝑋 ≠ 𝑌, then 𝐶 and 𝐷 are
disjoint.
(Contributed by Mario Carneiro, 14-Nov-2016.)
|
⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
& ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) ⇒ ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ 𝑋 ≠ 𝑌) → (𝐶 ∩ 𝐷) = ∅) |
|
Theorem | invdisj 3981* |
If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all
𝑦
∈ 𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥 ∈ 𝐴 are
disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
|
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝑥 → Disj 𝑥 ∈ 𝐴 𝐵) |
|
Theorem | disjiun 3982* |
A disjoint collection yields disjoint indexed unions for disjoint index
sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario
Carneiro, 14-Nov-2016.)
|
⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪
𝑥 ∈ 𝐷 𝐵) = ∅) |
|
Theorem | sndisj 3983 |
Any collection of singletons is disjoint. (Contributed by Mario
Carneiro, 14-Nov-2016.)
|
⊢ Disj 𝑥 ∈ 𝐴 {𝑥} |
|
Theorem | 0disj 3984 |
Any collection of empty sets is disjoint. (Contributed by Mario Carneiro,
14-Nov-2016.)
|
⊢ Disj 𝑥 ∈ 𝐴 ∅ |
|
Theorem | disjxsn 3985* |
A singleton collection is disjoint. (Contributed by Mario Carneiro,
14-Nov-2016.)
|
⊢ Disj 𝑥 ∈ {𝐴}𝐵 |
|
Theorem | disjx0 3986 |
An empty collection is disjoint. (Contributed by Mario Carneiro,
14-Nov-2016.)
|
⊢ Disj 𝑥 ∈ ∅ 𝐵 |
|
2.1.22 Binary relations
|
|
Syntax | wbr 3987 |
Extend wff notation to include the general binary relation predicate.
Note that the syntax is simply three class symbols in a row. Since binary
relations are the only possible wff expressions consisting of three class
expressions in a row, the syntax is unambiguous.
|
wff 𝐴𝑅𝐵 |
|
Definition | df-br 3988 |
Define a general binary relation. Note that the syntax is simply three
class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29
generalized to arbitrary classes. This definition of relations is
well-defined, although not very meaningful, when classes 𝐴 and/or
𝐵 are proper classes (i.e. are not
sets). On the other hand, we often
find uses for this definition when 𝑅 is a proper class (see for
example iprc 4877). (Contributed by NM, 31-Dec-1993.)
|
⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) |
|
Theorem | breq 3989 |
Equality theorem for binary relations. (Contributed by NM,
4-Jun-1995.)
|
⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)) |
|
Theorem | breq1 3990 |
Equality theorem for a binary relation. (Contributed by NM,
31-Dec-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) |
|
Theorem | breq2 3991 |
Equality theorem for a binary relation. (Contributed by NM,
31-Dec-1993.)
|
⊢ (𝐴 = 𝐵 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)) |
|
Theorem | breq12 3992 |
Equality theorem for a binary relation. (Contributed by NM,
8-Feb-1996.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |
|
Theorem | breqi 3993 |
Equality inference for binary relations. (Contributed by NM,
19-Feb-2005.)
|
⊢ 𝑅 = 𝑆 ⇒ ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵) |
|
Theorem | breq1i 3994 |
Equality inference for a binary relation. (Contributed by NM,
8-Feb-1996.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶) |
|
Theorem | breq2i 3995 |
Equality inference for a binary relation. (Contributed by NM,
8-Feb-1996.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵) |
|
Theorem | breq12i 3996 |
Equality inference for a binary relation. (Contributed by NM,
8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
|
⊢ 𝐴 = 𝐵
& ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷) |
|
Theorem | breq1d 3997 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶)) |
|
Theorem | breqd 3998 |
Equality deduction for a binary relation. (Contributed by NM,
29-Oct-2011.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝐴𝐷 ↔ 𝐶𝐵𝐷)) |
|
Theorem | breq2d 3999 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵)) |
|
Theorem | breq12d 4000 |
Equality deduction for a binary relation. (Contributed by NM,
8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|
⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷)) |