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Theorem List for Metamath Proof Explorer - 38101-38200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsumsnd 38101* A sum of a singleton is the term. The deduction version of sumsn 14184. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑘𝐵)    &   𝑘𝜑    &   ((𝜑𝑘 = 𝑀) → 𝐴 = 𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)
 
Theoremevthf 38102* A version of evth 22510 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐹    &   𝑦𝐹    &   𝑥𝑋    &   𝑦𝑋    &   𝑥𝜑    &   𝑦𝜑    &   𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
 
Theoremcnfex 38103 The class of continuous functions between two topologies is a set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
 
Theoremfnchoice 38104* For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝐴 ∈ Fin → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))
 
Theoremrefsumcn 38105* A finite sum of continuous real functions, from a common topological space, is continuous. The class expression for B normally contains free variables k and x to index it. See fsumcn 22425 for the analogous theorem on continuous complex functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ Σ𝑘𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
 
Theoremrfcnpre2 38106 If 𝐹 is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real 𝐵, is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐵    &   𝑥𝐹    &   𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑋 = 𝐽    &   𝐴 = {𝑥𝑋 ∣ (𝐹𝑥) < 𝐵}    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴𝐽)
 
Theoremcncmpmax 38107* When the hypothesis for the extreme value theorem hold, then the sup of the range of the function belongs to the range, it is real and it an upper bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑇 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑇 ≠ ∅)       (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡𝑇 (𝐹𝑡) ≤ sup(ran 𝐹, ℝ, < )))
 
Theoremrfcnpre3 38108* If F is a continuous function with respect to the standard topology, then the preimage A of the values greater or equal than a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐴 = {𝑡𝑇𝐵 ≤ (𝐹𝑡)}    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴 ∈ (Clsd‘𝐽))
 
Theoremrfcnpre4 38109* If F is a continuous function with respect to the standard topology, then the preimage A of the values smaller or equal than a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐴 = {𝑡𝑇 ∣ (𝐹𝑡) ≤ 𝐵}    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐴 ∈ (Clsd‘𝐽))
 
Theoremsumpair 38110* Sum of two distinct complex values. The class expression for 𝐴 and 𝐵 normally contain free variable 𝑘 to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑘𝐷)    &   (𝜑𝑘𝐸)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑𝐸 ∈ ℂ)    &   (𝜑𝐴𝐵)    &   ((𝜑𝑘 = 𝐴) → 𝐶 = 𝐷)    &   ((𝜑𝑘 = 𝐵) → 𝐶 = 𝐸)       (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
 
Theoremrfcnnnub 38111* Given a real continuous function 𝐹 defined on a compact topological space, there is always a positive integer that is a strict upper bound of its range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   𝑇 = 𝐽    &   (𝜑𝑇 ≠ ∅)    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐹𝐶)       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < 𝑛)
 
Theoremrefsum2cnlem1 38112* This is the core Lemma for refsum2cn 38113: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴    &   𝑥𝐹    &   𝑥𝐺    &   𝑥𝜑    &   𝐴 = (𝑘 ∈ {1, 2} ↦ if(𝑘 = 1, 𝐹, 𝐺))    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ ((𝐹𝑥) + (𝐺𝑥))) ∈ (𝐽 Cn 𝐾))
 
Theoremrefsum2cn 38113* The sum of two continuus real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐹    &   𝑥𝐺    &   𝑥𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ ((𝐹𝑥) + (𝐺𝑥))) ∈ (𝐽 Cn 𝐾))
 
Theoremelunnel2 38114 A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ (𝐵𝐶) ∧ ¬ 𝐴𝐶) → 𝐴𝐵)
 
Theoremadantlllr 38115 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (((((𝜑𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theorem3adantlr3 38116 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (((𝜑 ∧ (𝜓𝜒𝜂)) ∧ 𝜃) → 𝜏)
 
Theoremnnxrd 38117 A natural number is an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ*)
 
Theorem3adantll2 38118 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theorem3adantll3 38119 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜏)
 
Theoremssnel 38120 If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)
 
Theoremjcn 38121 Inference joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → ¬ (𝜓𝜒))
 
Theoremelabrexg 38122* Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
 
Theoremunicntop 38123 The underlying set of the standard topology on the complex numers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
ℂ = (TopOpen‘ℂfld)
 
Theoremifeq123d 38124 Equality deduction for conditional operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) AV: This theorem already exists as ifbieq12d 3966. TODO (NM): Please replace the usage of this theorem by ifbieq12d 3966 then delete this theorem. (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐷))
 
Theoremsncldre 38125 A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ℝ → {𝐴} ∈ (Clsd‘(topGen‘ran (,))))
 
Theoremcnopn 38126 The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
ℂ ∈ (TopOpen‘ℂfld)
 
Theoremn0p 38127 A polynomial with a nonzero coefficient is not the zero polynomial. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
((𝑃 ∈ (Poly‘ℤ) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘𝑃)‘𝑁) ≠ 0) → 𝑃 ≠ 0𝑝)
 
Theorempm2.65ni 38128 Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝜑𝜓)    &   𝜑 → ¬ 𝜓)       𝜑
 
Theoremelini 38129 Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐴𝐵    &   𝐴𝐶       𝐴 ∈ (𝐵𝐶)
 
Theorempwssfi 38130 Every element of the power set of 𝐴 is finite if and only if 𝐴 is finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴𝑉 → (𝐴 ∈ Fin ↔ 𝒫 𝐴 ⊆ Fin))
 
Theoremiuneq2df 38131 Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
 
Theoremnnfoctb 38132* There exists a mapping from onto any (nonempty) countable set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto𝐴)
 
Theoremssinss1d 38133 Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)
 
Theorem0un 38134 The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∪ 𝐴) = 𝐴
 
Theoremelpwinss 38135 An element of the powerset of 𝐵 intersected with anything, is a subset of 𝐵. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴 ∈ (𝒫 𝐵𝐶) → 𝐴𝐵)
 
Theoremunidmex 38136 If 𝐹 is a set, then dom 𝐹 is a set (common case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐹𝑉)    &   𝑋 = dom 𝐹       (𝜑𝑋 ∈ V)
 
Theoremndisj2 38137* A non disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑥 = 𝑦𝐵 = 𝐶)       Disj 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))
 
Theoremzenom 38138 The set of integer numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ℤ ≈ ω
 
Theoremrexsngf 38139* Restricted existential quantification over a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
 
Theoremuzwo4 38140* Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑗𝜓    &   (𝑗 = 𝑘 → (𝜑𝜓))       ((𝑆 ⊆ (ℤ𝑀) ∧ ∃𝑗𝑆 𝜑) → ∃𝑗𝑆 (𝜑 ∧ ∀𝑘𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))
 
Theoremunisn0 38141 The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
{∅} = ∅
 
Theoremssin0 38142 If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) = ∅)
 
Theoreminabs3 38143 Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴𝐶))
 
Theorempwpwuni 38144 Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵))
 
Theoremdisjiun2 38145* In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑Disj 𝑥𝐴 𝐵)    &   (𝜑𝐶𝐴)    &   (𝜑𝐷 ∈ (𝐴𝐶))    &   (𝑥 = 𝐷𝐵 = 𝐸)       (𝜑 → ( 𝑥𝐶 𝐵𝐸) = ∅)
 
Theorem0pwfi 38146 The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
∅ ∈ (𝒫 𝐴 ∩ Fin)
 
Theoremssinss2d 38147 Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐵𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)
 
Theoremzct 38148 The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ℤ ≼ ω
 
Theoremiunxsngf2 38149* A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theorempwfin0 38150 A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝒫 𝐴 ∩ Fin) ≠ ∅
 
Theoremuzct 38151 An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑍 = (ℤ𝑁)       𝑍 ≼ ω
 
Theoremiunxsnf 38152* A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝐶    &   𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ {𝐴}𝐵 = 𝐶
 
Theoremfiiuncl 38153* If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐷)    &   ((𝜑𝑦𝐷𝑧𝐷) → (𝑦𝑧) ∈ 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)       (𝜑 𝑥𝐴 𝐵𝐷)
 
Theoremiunp1 38154* The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑘𝐵    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)       (𝜑 𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = ( 𝑘 ∈ (𝑀...𝑁)𝐴𝐵))
 
Theoremfiunicl 38155* If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
((𝜑𝑥𝐴𝑦𝐴) → (𝑥𝑦) ∈ 𝐴)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ≠ ∅)       (𝜑 𝐴𝐴)
 
Theoremixpeq2d 38156 Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)
 
Theoremdisjxp1 38157* The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑Disj 𝑥𝐴 𝐵)       (𝜑Disj 𝑥𝐴 (𝐵 × 𝐶))
 
Theoremelpwd 38158 Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)
 
Theoremdisjsnxp 38159* The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Disj 𝑗𝐴 ({𝑗} × 𝐵)
 
Theoremeliind 38160* Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 𝑥𝐵 𝐶)    &   (𝜑𝐾𝐵)    &   (𝑥 = 𝐾 → (𝐴𝐶𝐴𝐷))       (𝜑𝐴𝐷)
 
Theoremrspcef 38161 Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑥𝜓    &   𝑥𝐴    &   𝑥𝐵    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
Theoreminn0f 38162 A non-empty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑥𝐴    &   𝑥𝐵       ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)
 
Theoremixpssmapc 38163* An infinite Cartesian product is a subset of set exponentiation. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑥𝜑    &   (𝜑𝐶𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑X𝑥𝐴 𝐵 ⊆ (𝐶𝑚 𝐴))
 
Theoreminn0 38164* A non-empty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)
 
Theoremelintd 38165* Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)
 
Theoremeqneltri 38166 If a class is not an element of another class, an equal class is also not an element. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝐴 = 𝐵    &    ¬ 𝐵𝐶        ¬ 𝐴𝐶
 
Theoremssdf 38167* A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑𝐴𝐵)
 
Theorembrneqtrd 38168 Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑 → ¬ 𝐴𝑅𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑 → ¬ 𝐴𝑅𝐶)
 
Theoremssnct 38169 A set containing an uncountable set is itself uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑 → ¬ 𝐴 ≼ ω)    &   (𝜑𝐴𝐵)       (𝜑 → ¬ 𝐵 ≼ ω)
 
Theoremssuniint 38170* Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)
 
Theoremelintdv 38171* Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)
 
Theoremssd 38172* A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑𝐴𝐵)
 
Theoremralimralim 38173 Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓𝜑))
 
Theoremsnelmap 38174 Membership of the element in the range of a constant map. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → (𝐴 × {𝑥}) ∈ (𝐵𝑚 𝐴))       (𝜑𝑥𝐵)
 
Theoremdfcleqf 38175 Equality connective between classes. Same as dfcleq 2508, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 
Theoremxrnmnfpnf 38176 An extended real that is neither real nor minus infinity, is plus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑 → ¬ 𝐴 ∈ ℝ)    &   (𝜑𝐴 ≠ -∞)       (𝜑𝐴 = +∞)
 
Theoremnelrnmpt 38177* Non-membership in the range of a function in maps-to notaion. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)       (𝜑 → ¬ 𝐶 ∈ ran 𝐹)
 
Theoremsnn0d 38178 The singleton of a set is not empty. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)       (𝜑 → {𝐴} ≠ ∅)
 
Theoremrabid3 38179 Membership in a restricted abstraction (special case). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐴 = {𝑥𝐵𝜑}       (𝑥𝐴 ↔ (𝑥𝐵𝜑))
 
Theoremiuneq1i 38180* Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐴 = 𝐵        𝑥𝐴 𝐶 = 𝑥𝐵 𝐶
 
Theoremnssrex 38181* Negation of subclass relationship. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)
 
Theoremnelpr2 38182 If a class is not an element of an unordered pair, it is not the second listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})       (𝜑𝐴𝐶)
 
Theoremnelpr1 38183 If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)    &   (𝜑 → ¬ 𝐴 ∈ {𝐵, 𝐶})       (𝜑𝐴𝐵)
 
Theoremiunssf 38184 Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐶       ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
 
Theoremelpwi2 38185 Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐵𝑉    &   𝐴𝐵       𝐴 ∈ 𝒫 𝐵
 
Theoremssinc 38186* Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑚 ∈ (𝑀..^𝑁)) → (𝐹𝑚) ⊆ (𝐹‘(𝑚 + 1)))       (𝜑 → (𝐹𝑀) ⊆ (𝐹𝑁))
 
Theoremssdec 38187* Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑚 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑚 + 1)) ⊆ (𝐹𝑚))       (𝜑 → (𝐹𝑁) ⊆ (𝐹𝑀))
 
Theoremelixpconstg 38188* Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝐹𝑉 → (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵))
 
Theoremiineq1d 38189* Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)
 
Theoremmetpsmet 38190 A metric is a pseudometric. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))
 
Theoremixpssixp 38191 Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
 
Theoremballss3 38192* A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   (𝜑𝐷 ∈ (PsMet‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑥𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥𝐴)       (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)
 
Theoremiunssd 38193* Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 𝑥𝐴 𝐵𝐶)
 
Theoremiunincfi 38194* Given a sequence of increasing sets, the union of a finite subsequence, is its last element. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹𝑛) ⊆ (𝐹‘(𝑛 + 1)))       (𝜑 𝑛 ∈ (𝑀...𝑁)(𝐹𝑛) = (𝐹𝑁))
 
Theoremnsstr 38195 If it's not a subclass, it's not a subclass of a smaller one. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
((¬ 𝐴𝐵𝐶𝐵) → ¬ 𝐴𝐶)
 
Theoremrabbida 38196 Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
Theoremrexanuz3 38197* Combine two different upper integer properties into one, for a single integer. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑗𝜑    &   𝑍 = (ℤ𝑀)    &   (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜒)    &   (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)𝜓)    &   (𝑘 = 𝑗 → (𝜒𝜃))    &   (𝑘 = 𝑗 → (𝜓𝜏))       (𝜑 → ∃𝑗𝑍 (𝜃𝜏))
 
Theoremrabeqd 38198* Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})
 
Theoremcbvmpt22 38199* Rule to change the second bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑦𝐴    &   𝑤𝐴    &   𝑤𝐶    &   𝑦𝐸    &   (𝑦 = 𝑤𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑤𝐵𝐸)
 
Theoremcbvmpt21 38200* Rule to change the first bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐵    &   𝑧𝐵    &   𝑧𝐶    &   𝑥𝐸    &   (𝑥 = 𝑧𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑦𝐵𝐸)
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