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Theorem List for Metamath Proof Explorer - 31701-31800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremor3di 31701 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
((𝜑 ∨ (𝜓𝜒𝜏)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒) ∧ (𝜑𝜏)))
 
Theoremor3dir 31702 Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
(((𝜑𝜓𝜒) ∨ 𝜏) ↔ ((𝜑𝜏) ∧ (𝜓𝜏) ∧ (𝜒𝜏)))
 
Theorem3o1cs 31703 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
Theorem3o2cs 31704 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜓𝜃)
 
Theorem3o3cs 31705 Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
((𝜑𝜓𝜒) → 𝜃)       (𝜒𝜃)
 
Theorem13an22anass 31706 Associative law for four conjunctions with a triple conjunction. (Contributed by Thierry Arnoux, 21-Jan-2025.)
((𝜑 ∧ (𝜓𝜒𝜃)) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
 
21.3.2  Predicate Calculus
 
21.3.2.1  Predicate Calculus - misc additions
 
Theoremsbc2iedf 31707* Conversion of implicit substitution to explicit class substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝜒    &   𝑦𝜒    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
 
Theoremrspc2daf 31708* Double restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝜒    &   𝑦𝜒    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))    &   (𝜑 → ∀𝑥𝑉𝑦𝑊 𝜓)       (𝜑𝜒)
 
21.3.2.2  Restricted quantification - misc additions
 
Theoremralcom4f 31709* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
𝑦𝐴       (∀𝑥𝐴𝑦𝜑 ↔ ∀𝑦𝑥𝐴 𝜑)
 
Theoremrexcom4f 31710* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
𝑦𝐴       (∃𝑥𝐴𝑦𝜑 ↔ ∃𝑦𝑥𝐴 𝜑)
 
Theorem19.9d2rf 31711 A deduction version of one direction of 19.9 2199 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜓)
 
Theorem19.9d2r 31712* A deduction version of one direction of 19.9 2199 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)       (𝜑𝜓)
 
Theoremr19.29ffa 31713* A commonly used pattern based on r19.29 3115, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
((((𝜑𝑥𝐴) ∧ 𝑦𝐵) ∧ 𝜓) → 𝜒)       ((𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → 𝜒)
 
21.3.2.3  Equality
 
Theoremeqtrb 31714 A transposition of equality. (Contributed by Thierry Arnoux, 20-Aug-2023.)
((𝐴 = 𝐵𝐴 = 𝐶) ↔ (𝐴 = 𝐵𝐵 = 𝐶))
 
Theoremeqelbid 31715* A variable elimination law for equality within a given set 𝐴. See equvel 2456. (Contributed by Thierry Arnoux, 20-Feb-2025.)
(𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)       (𝜑 → (∀𝑥𝐴 (𝑥 = 𝐵𝑥 = 𝐶) ↔ 𝐵 = 𝐶))
 
21.3.2.4  Double restricted existential uniqueness quantification
 
Theoremopsbc2ie 31716* Conversion of implicit substitution to explicit class substitution for ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))       (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑[𝑦 / 𝑏][𝑥 / 𝑎]𝜒))
 
Theoremopreu2reuALT 31717* Correspondence between uniqueness of ordered pairs and double restricted existential uniqueness quantification. Alternate proof of one direction only, use opreu2reurex 6294 instead. (Contributed by Thierry Arnoux, 4-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))       ((∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒) → ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑)
 
21.3.2.5  Double restricted existential uniqueness quantification syntax
 
Syntaxw2reu 31718 Syntax for double restricted existential uniqueness quantification.
wff ∃!𝑥𝐴 , 𝑦𝐵𝜑
 
Definitiondf-2reu 31719 Define the double restricted existential uniqueness quantifier. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑))
 
Theorem2reucom 31720 Double restricted existential uniqueness commutes. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ ∃!𝑦𝐵 , 𝑥𝐴𝜑)
 
Theorem2reu2rex1 31721 Double restricted existential uniqueness implies double restricted existence. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)
 
Theorem2reureurex 31722 Double restricted existential uniqueness implies restricted existential uniqueness with restricted existence. (Contributed by AV, 5-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃!𝑥𝐴𝑦𝐵 𝜑)
 
Theorem2reu2reu2 31723* Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023.)
(∃!𝑥𝐴 , 𝑦𝐵𝜑 → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
 
Theoremopreu2reu1 31724* Equivalent definition of the double restricted existential uniqueness quantifier, using uniqueness of ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)
(𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒𝜑))       (∃!𝑥𝐴 , 𝑦𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)
 
Theoremsq2reunnltb 31725* There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4𝑘 + 1. Double restricted existential uniqueness variant of 2sqreunnltb 26964. (Contributed by AV, 5-Jul-2023.)
(𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ , 𝑏 ∈ ℕ(𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
 
Theoremaddsqnot2reu 31726* For each complex number 𝐶, there does not uniquely exist two complex numbers 𝑎 and 𝑏, with 𝑏 squared and added to 𝑎 resulting in the given complex number 𝐶. Double restricted existential uniqueness variant of addsqn2reurex2 26948. (Contributed by AV, 5-Jul-2023.)
(𝐶 ∈ ℂ → ¬ ∃!𝑎 ∈ ℂ , 𝑏 ∈ ℂ(𝑎 + (𝑏↑2)) = 𝐶)
 
21.3.2.6  Substitution (without distinct variables) - misc additions
 
Theoremsbceqbidf 31727 Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
𝑥𝜑    &   (𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))
 
Theoremsbcies 31728* A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
𝐴 = (𝐸𝑊)    &   (𝑎 = 𝐴 → (𝜑𝜓))       (𝑤 = 𝑊 → ([(𝐸𝑤) / 𝑎]𝜓𝜑))
 
21.3.2.7  Existential "at most one" - misc additions
 
Theoremmo5f 31729* Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
𝑖𝜑    &   𝑗𝜑       (∃*𝑥𝜑 ↔ ∀𝑖𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
 
Theoremnmo 31730* Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
𝑦𝜑       (¬ ∃*𝑥𝜑 ↔ ∀𝑦𝑥(𝜑𝑥𝑦))
 
21.3.2.8  Existential uniqueness - misc additions
 
Theoremreuxfrdf 31731* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 3745 (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) (Revised by Thierry Arnoux, 30-Mar-2018.)
𝑦𝐵    &   ((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃*𝑦𝐶 𝑥 = 𝐴)       (𝜑 → (∃!𝑥𝐵𝑦𝐶 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐶 𝜓))
 
Theoremrexunirn 31732* Restricted existential quantification over the union of the range of a function. Cf. rexrn 7089 and eluni2 4913. (Contributed by Thierry Arnoux, 19-Sep-2017.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑥𝐴𝐵𝑉)       (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦 ran 𝐹𝜑)
 
21.3.2.9  Restricted "at most one" - misc additions
 
Theoremrmoxfrd 31733* Transfer "at most one" restricted quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐵 𝜓 ↔ ∃*𝑦𝐶 𝜒))
 
Theoremrmoun 31734 "At most one" restricted existential quantifier for a union implies the same quantifier on both sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
(∃*𝑥 ∈ (𝐴𝐵)𝜑 → (∃*𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑))
 
Theoremrmounid 31735* A case where an "at most one" restricted existential quantifier for a union is equivalent to such a quantifier for one of the sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
((𝜑𝑥𝐵) → ¬ 𝜓)       (𝜑 → (∃*𝑥 ∈ (𝐴𝐵)𝜓 ↔ ∃*𝑥𝐴 𝜓))
 
21.3.2.10  Restricted iota (description binder)
 
Theoremriotaeqbidva 31736* Equivalent wff's yield equal restricted definition binders (deduction form). (raleqbidva 3328 analog.) (Contributed by Thierry Arnoux, 29-Jan-2025.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜒))
 
21.3.3  General Set Theory
 
21.3.3.1  Class abstractions (a.k.a. class builders)
 
Theoremdmrab 31737* Domain of a restricted class abstraction over a cartesian product. (Contributed by Thierry Arnoux, 3-Jul-2023.)
(𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))       dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥𝐴 ∣ ∃𝑦𝐵 𝜓}
 
Theoremdifrab2 31738 Difference of two restricted class abstractions. Compare with difrab 4309. (Contributed by Thierry Arnoux, 3-Jan-2022.)
({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
 
TheoremrabexgfGS 31739 Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)
𝑥𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 
Theoremrabsnel 31740* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)
𝐵 ∈ V       ({𝑥𝐴𝜑} = {𝐵} → 𝐵𝐴)
 
Theoremeqrrabd 31741* Deduce equality with a restricted abstraction. (Contributed by Thierry Arnoux, 11-Apr-2024.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐴) → (𝑥𝐵𝜓))       (𝜑𝐵 = {𝑥𝐴𝜓})
 
Theoremforesf1o 31742* From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝐴𝑉𝐹:𝐴onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹𝑥):𝑥1-1-onto𝐵)
 
Theoremrabfodom 31743* Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝜑𝑥𝐴𝑦 = (𝐹𝑥)) → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴onto𝐵)       (𝜑 → {𝑦𝐵𝜒} ≼ {𝑥𝐴𝜓})
 
21.3.3.2  Image Sets
 
Theoremabrexdomjm 31744* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑦𝐴 → ∃*𝑥𝜑)       (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)
 
Theoremabrexdom2jm 31745* An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ≼ 𝐴)
 
Theoremabrexexd 31746* Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
𝑥𝐴    &   (𝜑𝐴 ∈ V)       (𝜑 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
 
Theoremelabreximd 31747* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
𝑥𝜑    &   𝑥𝜒    &   (𝐴 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐶) → 𝜓)       ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)
 
Theoremelabreximdv 31748* Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
(𝐴 = 𝐵 → (𝜒𝜓))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐶) → 𝜓)       ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)
 
Theoremabrexss 31749* A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
𝑥𝐶       (∀𝑥𝐴 𝐵𝐶 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶)
 
21.3.3.3  Set relations and operations - misc additions
 
Theoremelunsn 31750 Elementhood to a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.)
(𝐴𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 = 𝐶)))
 
Theoremnelun 31751 Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝐴 = (𝐵𝐶) → (¬ 𝑋𝐴 ↔ (¬ 𝑋𝐵 ∧ ¬ 𝑋𝐶)))
 
Theoremsnsssng 31752 If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.) (Revised by Thierry Arnoux, 11-Apr-2024.)
((𝐴𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)
 
Theoreminin 31753 Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
 
Theoreminindif 31754 See inundif 4479. (Contributed by Thierry Arnoux, 13-Sep-2017.)
((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
 
Theoremdifininv 31755 Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021.)
((((𝐴𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶𝐴) ∩ 𝐵) = ∅) → (𝐴𝐵) = (𝐶𝐵))
 
Theoremdifeq 31756 Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴𝐵) = 𝐶 ↔ ((𝐶𝐵) = ∅ ∧ (𝐶𝐵) = (𝐴𝐵)))
 
Theoremeqdif 31757 If both set differences of two sets are empty, those sets are equal. (Contributed by Thierry Arnoux, 16-Nov-2023.)
(((𝐴𝐵) = ∅ ∧ (𝐵𝐴) = ∅) → 𝐴 = 𝐵)
 
Theoremindifbi 31758 Two ways to express equality relative to a class 𝐴. (Contributed by Thierry Arnoux, 23-Jun-2024.)
((𝐴𝐵) = (𝐴𝐶) ↔ (𝐴𝐵) = (𝐴𝐶))
 
Theoremdiffib 31759 Case where diffi 9179 is a biconditional. (Contributed by Thierry Arnoux, 27-Jun-2024.)
(𝐵 ∈ Fin → (𝐴 ∈ Fin ↔ (𝐴𝐵) ∈ Fin))
 
Theoremdifxp1ss 31760 Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
((𝐴𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)
 
Theoremdifxp2ss 31761 Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
(𝐴 × (𝐵𝐶)) ⊆ (𝐴 × 𝐵)
 
Theoremindifundif 31762 A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
(((𝐴𝐵) ∖ 𝐶) ∪ (𝐴𝐵)) = (𝐴 ∖ (𝐵𝐶))
 
Theoremelpwincl1 31763 Closure of intersection with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐴 ∈ 𝒫 𝐶)       (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
 
Theoremelpwdifcl 31764 Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐴 ∈ 𝒫 𝐶)       (𝜑 → (𝐴𝐵) ∈ 𝒫 𝐶)
 
Theoremelpwiuncl 31765* Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ 𝒫 𝐶)       (𝜑 𝑘𝐴 𝐵 ∈ 𝒫 𝐶)
 
21.3.3.4  Unordered pairs
 
Theoremeqsnd 31766* Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.)
((𝜑𝑥𝐴) → 𝑥 = 𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = {𝐵})
 
Theoremelpreq 31767 Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
(𝜑𝑋 ∈ {𝐴, 𝐵})    &   (𝜑𝑌 ∈ {𝐴, 𝐵})    &   (𝜑 → (𝑋 = 𝐴𝑌 = 𝐴))       (𝜑𝑋 = 𝑌)
 
Theoremnelpr 31768 A set 𝐴 not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝐴𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴𝐵𝐴𝐶)))
 
Theoreminpr0 31769 Rewrite an empty intersection with a pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵𝐴 ∧ ¬ 𝐶𝐴))
 
Theoremneldifpr1 31770 The first element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})
 
Theoremneldifpr2 31771 The second element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})
 
Theoremunidifsnel 31772 The other element of a pair is an element of the pair. (Contributed by Thierry Arnoux, 26-Aug-2017.)
((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ∈ 𝑃)
 
Theoremunidifsnne 31773 The other element of a pair is not the known element. (Contributed by Thierry Arnoux, 26-Aug-2017.)
((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ {𝑋}) ≠ 𝑋)
 
21.3.3.5  Conditional operator - misc additions
 
Theoremifeqeqx 31774* An equality theorem tailored for ballotlemsf1o 33512. (Contributed by Thierry Arnoux, 14-Apr-2017.)
(𝑥 = 𝑋𝐴 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝑎)    &   (𝑥 = 𝑋 → (𝜒𝜃))    &   (𝑥 = 𝑌 → (𝜒𝜓))    &   (𝜑𝑎 = 𝐶)    &   ((𝜑𝜓) → 𝜃)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑊)       ((𝜑𝑥 = if(𝜓, 𝑋, 𝑌)) → 𝑎 = if(𝜒, 𝐴, 𝐵))
 
Theoremelimifd 31775 Elimination of a conditional operator contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐴 → (𝜒𝜃)))    &   (𝜑 → (if(𝜓, 𝐴, 𝐵) = 𝐵 → (𝜒𝜏)))       (𝜑 → (𝜒 ↔ ((𝜓𝜃) ∨ (¬ 𝜓𝜏))))
 
Theoremelim2if 31776 Elimination of two conditional operators contained in a wff 𝜒. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))       (𝜒 ↔ ((𝜑𝜃) ∨ (¬ 𝜑 ∧ ((𝜓𝜏) ∨ (¬ 𝜓𝜂)))))
 
Theoremelim2ifim 31777 Elimination of two conditional operators for an implication. (Contributed by Thierry Arnoux, 25-Jan-2017.)
(if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐴 → (𝜒𝜃))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐵 → (𝜒𝜏))    &   (if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = 𝐶 → (𝜒𝜂))    &   (𝜑𝜃)    &   ((¬ 𝜑𝜓) → 𝜏)    &   ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜂)       𝜒
 
Theoremifeq3da 31778 Given an expression 𝐶 containing if(𝜓, 𝐸, 𝐹), substitute (hypotheses .1 and .2) and evaluate (hypotheses .3 and .4) it for both cases at the same time. (Contributed by Thierry Arnoux, 13-Dec-2021.)
(if(𝜓, 𝐸, 𝐹) = 𝐸𝐶 = 𝐺)    &   (if(𝜓, 𝐸, 𝐹) = 𝐹𝐶 = 𝐻)    &   (𝜑𝐺 = 𝐴)    &   (𝜑𝐻 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)
 
Theoremifnetrue 31779 Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
 
Theoremifnefals 31780 Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)
 
Theoremifnebib 31781 The converse of ifbi 4551 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
(𝐴𝐵 → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑𝜓)))
 
21.3.3.6  Set union
 
Theoremuniinn0 31782* Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)
(( 𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 (𝑥𝐵) ≠ ∅)
 
Theoremuniin1 31783* Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
𝑥𝐴 (𝑥𝐵) = ( 𝐴𝐵)
 
Theoremuniin2 31784* Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)
𝑥𝐵 (𝐴𝑥) = (𝐴 𝐵)
 
Theoremdifuncomp 31785 Express a class difference using unions and class complements. (Contributed by Thierry Arnoux, 21-Jun-2020.)
(𝐴𝐶 → (𝐴𝐵) = (𝐶 ∖ ((𝐶𝐴) ∪ 𝐵)))
 
Theoremelpwunicl 31786 Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.) (Proof shortened by BJ, 6-Apr-2024.)
(𝜑𝐴 ∈ 𝒫 𝒫 𝐵)       (𝜑 𝐴 ∈ 𝒫 𝐵)
 
21.3.3.7  Indexed union - misc additions
 
Theoremcbviunf 31787* 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.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
Theoremiuneq12daf 31788 Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑥𝐴) → 𝐶 = 𝐷)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐷)
 
Theoremiunin1f 31789 Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 5062 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.) (Revised by Thierry Arnoux, 2-May-2020.)
𝑥𝐶        𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵𝐶)
 
Theoremssiun3 31790* Subset equivalence for an indexed union. (Contributed by Thierry Arnoux, 17-Oct-2016.)
(∀𝑦𝐶𝑥𝐴 𝑦𝐵𝐶 𝑥𝐴 𝐵)
 
Theoremssiun2sf 31791 Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
𝑥𝐴    &   𝑥𝐶    &   𝑥𝐷    &   (𝑥 = 𝐶𝐵 = 𝐷)       (𝐶𝐴𝐷 𝑥𝐴 𝐵)
 
Theoremiuninc 31792* The union of an increasing collection of sets is its last element. (Contributed by Thierry Arnoux, 22-Jan-2017.)
(𝜑𝐹 Fn ℕ)    &   ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))       ((𝜑𝑖 ∈ ℕ) → 𝑛 ∈ (1...𝑖)(𝐹𝑛) = (𝐹𝑖))
 
Theoremiundifdifd 31793* The intersection of a set is the complement of the union of the complements. (Contributed by Thierry Arnoux, 19-Dec-2016.)
(𝐴 ⊆ 𝒫 𝑂 → (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥))))
 
Theoremiundifdif 31794* The intersection of a set is the complement of the union of the complements. TODO: shorten using iundifdifd 31793. (Contributed by Thierry Arnoux, 4-Sep-2016.)
𝑂 ∈ V    &   𝐴 ⊆ 𝒫 𝑂       (𝐴 ≠ ∅ → 𝐴 = (𝑂 𝑥𝐴 (𝑂𝑥)))
 
Theoremiunrdx 31795* Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
(𝜑𝐹:𝐴onto𝐶)    &   ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)       (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
 
Theoremiunpreima 31796* Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
(Fun 𝐹 → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
 
Theoremiunrnmptss 31797* A subset relation for an indexed union over the range of function expressed as a mapping. (Contributed by Thierry Arnoux, 27-Mar-2018.)
(𝑦 = 𝐵𝐶 = 𝐷)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 𝑥𝐴 𝐷)
 
Theoremiunxunsn 31798* Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋𝐵 = 𝐶)       (𝑋𝑉 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = ( 𝑥𝐴 𝐵𝐶))
 
Theoremiunxunpr 31799* Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
 
21.3.3.8  Indexed intersection - misc additions
 
Theoremiinabrex 31800* Rewriting an indexed intersection into an intersection of its image set. (Contributed by Thierry Arnoux, 15-Jun-2024.)
(∀𝑥𝐴 𝐵𝑉 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
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