P. 51 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5001-5100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iunss2 5001*	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 4894. (Contributed by NM, 9-Dec-2004.)
⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷)
Theorem	iunssd 5002*	Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
Theorem	iunab 5003*	The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)
⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}
Theorem	iunrab 5004*	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 5005*	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 5006	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 5007*	Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
Theorem	iinss 5008*	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 5009	An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)
Theorem	uniiun 5010*	Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
Theorem	intiin 5011*	Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
Theorem	iunid 5012*	An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) (Proof shortened by SN, 15-Jan-2025.)
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
Theorem	iunidOLD 5013*	Obsolete version of iunid 5012 as of 15-Jan-2025. (Contributed by NM, 6-Sep-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
Theorem	iun0 5014	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 5015	An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ∪ 𝑥 ∈ ∅ 𝐴 = ∅
Theorem	0iin 5016	An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
⊢ ∩ 𝑥 ∈ ∅ 𝐴 = V
Theorem	viin 5017*	Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2203 and abid2 2865. When 𝐴 = 𝑥, this evaluates to ∅ by intiin 5011 and intv 5306. (Contributed by NM, 11-Sep-2008.)
⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
Theorem	iunsn 5018*	Indexed union of a singleton. Compare dfiun2 4985 and rnmpt 5903. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ ∪ 𝑥 ∈ 𝐴 {𝐵} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
Theorem	iunn0 5019*	There is a nonempty class in an indexed collection 𝐵(𝑥) iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅)
Theorem	iinab 5020*	Indexed intersection of a class abstraction. (Contributed by NM, 6-Dec-2011.)
⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}
Theorem	iinrab 5021*	Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑})
Theorem	iinrab2 5022*	Indexed intersection of a restricted class abstraction. (Contributed by NM, 6-Dec-2011.)
⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝜑} ∩ 𝐵) = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐴 𝜑}
Theorem	iunin2 5023*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5010 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶)
Theorem	iunin1 5024*	Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5010 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵)
Theorem	iinun2 5025*	Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 5011 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
⊢ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶)
Theorem	iundif2 5026*	Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 5011 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶)
Theorem	iindif1 5027*	Indexed intersection of class difference with the subtrahend held constant. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶))
Theorem	2iunin 5028*	Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)
Theorem	iindif2 5029*	Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 5010 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶))
Theorem	iinin2 5030*	Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 5011 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶))
Theorem	iinin1 5031*	Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 5011 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵))
Theorem	iinvdif 5032*	The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.)
⊢ ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	elriin 5033*	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 5034*	Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = 𝐴)
Theorem	riinn0 5035*	Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ ((∀𝑥 ∈ 𝑋 𝑆 ⊆ 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 𝑆) = ∩ 𝑥 ∈ 𝑋 𝑆)
Theorem	riinrab 5036*	Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)
⊢ (𝐴 ∩ ∩ 𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝜑}) = {𝑦 ∈ 𝐴 ∣ ∀𝑥 ∈ 𝑋 𝜑}
Theorem	symdif0 5037	Symmetric difference with the empty class. The empty class is the identity element for symmetric difference. (Contributed by Scott Fenton, 24-Apr-2012.)
⊢ (𝐴 △ ∅) = 𝐴
Theorem	symdifv 5038	The symmetric difference with the universal class is the complement. (Contributed by Scott Fenton, 24-Apr-2012.)
⊢ (𝐴 △ V) = (V ∖ 𝐴)
Theorem	symdifid 5039	The symmetric difference of a class with itself is the empty class. (Contributed by Scott Fenton, 25-Apr-2012.)
⊢ (𝐴 △ 𝐴) = ∅
Theorem	iinxsng 5040*	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 5041*	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 5042*	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 5043*	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 5044*	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.) Avoid ax-13 2370. (Revised by GG, 19-May-2023.)
⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Theorem	iunun 5045	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 5046	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	iunxdif3 5047*	An indexed union where some terms are the empty set. See iunxdif2 5005. (Contributed by Thierry Arnoux, 4-May-2020.)
⊢ Ⅎ𝑥𝐸    ⇒   ⊢ (∀𝑥 ∈ 𝐸 𝐵 = ∅ → ∪ 𝑥 ∈ (𝐴 ∖ 𝐸)𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
Theorem	iunxprg 5048*	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 5049*	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	iinuni 5050*	A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
⊢ (𝐴 ∪ ∩ 𝐵) = ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥)
Theorem	iununi 5051*	A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
⊢ ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥))
Theorem	sspwuni 5052	Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
Theorem	pwssb 5053*	Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
Theorem	elpwpw 5054	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 5055*	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 5056*	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 5057	Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵))
Theorem	iinpw 5058*	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 5059*	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	intss2 5060	A nonempty intersection of a family of subsets of a class is included in that class. (Contributed by BJ, 7-Dec-2021.)
⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋))
Theorem	rintn0 5061	Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ 𝑋 ≠ ∅) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋)
2.1.22  Disjointness
Syntax	wdisj 5062	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 5063*	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 5064*	Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
Theorem	disjss2 5065	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 5066	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 5067*	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 5068*	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 5069*	Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
Theorem	disjeq1d 5070*	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 5071*	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 5072*	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 5073*	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	nfdisjw 5074*	Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5075 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2370. (Revised by GG, 26-Jan-2024.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵
Theorem	nfdisj 5075	Bound-variable hypothesis builder for disjoint collection. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker nfdisjw 5074 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵
Theorem	nfdisj1 5076	Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵
Theorem	disjor 5077*	Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)    ⇒   ⊢ (Disj 𝑖 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (𝐵 ∩ 𝐶) = ∅))
Theorem	disjors 5078*	Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
Theorem	disji2 5079*	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	disji 5080*	Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑋 = 𝑌. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)    ⇒   ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝑍 ∈ 𝐶 ∧ 𝑍 ∈ 𝐷)) → 𝑋 = 𝑌)
Theorem	invdisj 5081*	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	invdisjrab 5082*	The restricted class abstractions {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦} for distinct 𝑦 ∈ 𝐴 are disjoint. (Contributed by AV, 6-May-2020.) (Proof shortened by GG, 26-Jan-2024.)
⊢ Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}
Theorem	disjiun 5083*	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	disjord 5084*	Conditions for a collection of sets 𝐴(𝑎) for 𝑎 ∈ 𝑉 to be disjoint. (Contributed by AV, 9-Jan-2022.)
⊢ (𝑎 = 𝑏 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑎 = 𝑏)    ⇒   ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 𝐴)
Theorem	disjiunb 5085*	Two ways to say that a collection of index unions 𝐶(𝑖, 𝑥) for 𝑖 ∈ 𝐴 and 𝑥 ∈ 𝐵 is disjoint. (Contributed by AV, 9-Jan-2022.)
⊢ (𝑖 = 𝑗 → 𝐵 = 𝐷)    &   ⊢ (𝑖 = 𝑗 → 𝐶 = 𝐸)    ⇒   ⊢ (Disj 𝑖 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (∪ 𝑥 ∈ 𝐵 𝐶 ∩ ∪ 𝑥 ∈ 𝐷 𝐸) = ∅))
Theorem	disjiund 5086*	Conditions for a collection of index unions of sets 𝐴(𝑎, 𝑏) for 𝑎 ∈ 𝑉 and 𝑏 ∈ 𝑊 to be disjoint. (Contributed by AV, 9-Jan-2022.)
⊢ (𝑎 = 𝑐 → 𝐴 = 𝐶)    &   ⊢ (𝑏 = 𝑑 → 𝐶 = 𝐷)    &   ⊢ (𝑎 = 𝑐 → 𝑊 = 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐷) → 𝑎 = 𝑐)    ⇒   ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ 𝑊 𝐴)
Theorem	sndisj 5087	Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ 𝐴 {𝑥}
Theorem	0disj 5088	Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ 𝐴 ∅
Theorem	disjxsn 5089*	A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ {𝐴}𝐵
Theorem	disjx0 5090	An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ Disj 𝑥 ∈ ∅ 𝐵
Theorem	disjprg 5091*	A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)    &   ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷 ∩ 𝐸) = ∅))
Theorem	disjxiun 5092*	An indexed union of a disjoint collection of disjoint collections is disjoint if each component is disjoint, and the disjoint unions in the collection are also disjoint. Note that 𝐵(𝑦) and 𝐶(𝑥) may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by JJ, 27-May-2021.)
⊢ (Disj 𝑦 ∈ 𝐴 𝐵 → (Disj 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 ↔ (∀𝑦 ∈ 𝐴 Disj 𝑥 ∈ 𝐵 𝐶 ∧ Disj 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶)))
Theorem	disjxun 5093*	The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐴 ∩ 𝐵) = ∅ → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ (Disj 𝑥 ∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)))
Theorem	disjss3 5094*	Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ (𝐵 ∖ 𝐴)𝐶 = ∅) → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶))
2.1.23  Binary relations
Syntax	wbr 5095	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. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 11366.)
wff 𝐴𝑅𝐵
Definition	df-br 5096	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. Class 𝑅 often denotes a relation such as "< " that compares two classes 𝐴 and 𝐵, which might be numbers such as 1 and 2 (see df-ltxr 11173 for the specific definition of <). As a wff, relations are true or false. For example, (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9) (ex-br 30393). Often class 𝑅 meets the Rel criteria to be defined in df-rel 5630, and in particular 𝑅 may be a function (see df-fun 6488). 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 7851). (Contributed by NM, 31-Dec-1993.)
⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
Theorem	breq 5097	Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
Theorem	breq1 5098	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
⊢ (𝐴 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐶))
Theorem	breq2 5099	Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
⊢ (𝐴 = 𝐵 → (𝐶𝑅𝐴 ↔ 𝐶𝑅𝐵))
Theorem	breq12 5100	Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))