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Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulltgt0 39001 The product of a negative and a positive number is negative. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 · 𝐵) < 0)

Theoremrspcegf 39002 A version of rspcev 3304 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝜓    &   𝑥𝐴    &   𝑥𝐵    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Theoremrabexgf 39003 A version of rabexg 4803 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑥𝐴       (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremfcnre 39004 A function continuous with respect to the standard topology, is a real mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐹𝐶)       (𝜑𝐹:𝑇⟶ℝ)

Theoremsumsnd 39005* A sum of a singleton is the term. The deduction version of sumsn 14456. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑘𝐵)    &   𝑘𝜑    &   ((𝜑𝑘 = 𝑀) → 𝐴 = 𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)

Theoremevthf 39006* A version of evth 22739 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 39007 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 39008* 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 39009* 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 22654 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 39010 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 39011* 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 39012* 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 39013* 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 39014* 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 39015* 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 39016* This is the core Lemma for refsum2cn 39017: 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 39017* 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 39018 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 39019 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       (((((𝜑𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem3adantlr3 39020 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)       (((𝜑 ∧ (𝜓𝜒𝜂)) ∧ 𝜃) → 𝜏)

Theoremnnxrd 39021 A natural number is an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℕ)       (𝜑𝐴 ∈ ℝ*)

Theorem3adantll2 39022 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((((𝜑𝜂𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theorem3adantll3 39023 Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)       ((((𝜑𝜓𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜏)

Theoremssnel 39024 If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴𝐵 ∧ ¬ 𝐶𝐵) → ¬ 𝐶𝐴)

Theoremjcn 39025 Inference joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝜓)    &   (𝜑 → ¬ 𝜒)       (𝜑 → ¬ (𝜓𝜒))

Theoremelabrexg 39026* Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝑥𝐴𝐵𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremifeq123d 39027 Equality deduction for conditional operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) AV: This theorem already exists as ifbieq12d 4104. TODO (NM): Please replace the usage of this theorem by ifbieq12d 4104 then delete this theorem. (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐷))

Theoremsncldre 39028 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 (,))))

Theoremn0p 39029 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 39030 Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝜑𝜓)    &   𝜑 → ¬ 𝜓)       𝜑

Theorempwssfi 39031 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 39032 Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)

Theoremnnfoctb 39033* 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 39034 Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)

Theorem0un 39035 The union of the empty set with a class is itself. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(∅ ∪ 𝐴) = 𝐴

Theoremelpwinss 39036 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 39037 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 39038* A non disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝑥 = 𝑦𝐵 = 𝐶)       Disj 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦𝐴 (𝑥𝑦 ∧ (𝐵𝐶) ≠ ∅))

Theoremzenom 39039 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 39040* Restricted existential quantification over a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))

Theoremuzwo4 39041* 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 39042 The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
{∅} = ∅

Theoremssin0 39043 If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐷) = ∅)

Theoreminabs3 39044 Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐶𝐵 → ((𝐴𝐵) ∩ 𝐶) = (𝐴𝐶))

Theorempwpwuni 39045 Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵))

Theoremdisjiun2 39046* 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 39047 The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
∅ ∈ (𝒫 𝐴 ∩ Fin)

Theoremssinss2d 39048 Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐵𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)

Theoremzct 39049 The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
ℤ ≼ ω

Theoremiunxsngf2 39050* 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 39051 A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝒫 𝐴 ∩ Fin) ≠ ∅

Theoremuzct 39052 An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝑍 = (ℤ𝑁)       𝑍 ≼ ω

Theoremiunxsnf 39053* 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 39054* 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 39055* 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 39056* 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 39057 Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑X𝑥𝐴 𝐵 = X𝑥𝐴 𝐶)

Theoremdisjxp1 39058* 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))

Theoremdisjsnxp 39059* 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 39060* Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴 𝑥𝐵 𝐶)    &   (𝜑𝐾𝐵)    &   (𝑥 = 𝐾 → (𝐴𝐶𝐴𝐷))       (𝜑𝐴𝐷)

Theoremrspcef 39061 Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑥𝜓    &   𝑥𝐴    &   𝑥𝐵    &   (𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Theoreminn0f 39062 A non-empty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
𝑥𝐴    &   𝑥𝐵       ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)

Theoremixpssmapc 39063* 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 39064* A non-empty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
((𝐴𝐵) ≠ ∅ ↔ ∃𝑥𝐴 𝑥𝐵)

Theoremelintd 39065* Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)

Theoremeqneltri 39066 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 39067* A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑𝐴𝐵)

Theorembrneqtrd 39068 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 39069 A set containing an uncountable set is itself uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑 → ¬ 𝐴 ≼ ω)    &   (𝜑𝐴𝐵)       (𝜑 → ¬ 𝐵 ≼ ω)

Theoremssuniint 39070* 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 39071* Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐵) → 𝐴𝑥)       (𝜑𝐴 𝐵)

Theoremssd 39072* A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑𝐴𝐵)

Theoremralimralim 39073 Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(∀𝑥𝐴 𝜑 → ∀𝑥𝐴 (𝜓𝜑))

Theoremsnelmap 39074 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 39075 Equality connective between classes. Same as dfcleq 2614, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐴    &   𝑥𝐵       (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))

Theoremxrnmnfpnf 39076 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 39077* 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 39078 The singleton of a set is not empty. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴𝑉)       (𝜑 → {𝐴} ≠ ∅)

Theoremiuneq1i 39079* Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐴 = 𝐵        𝑥𝐴 𝐶 = 𝑥𝐵 𝐶

Theoremnssrex 39080* Negation of subclass relationship. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)

Theoremnelpr2 39081 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 39082 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 39083 Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐶       ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)

Theoremssinc 39084* 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 39085* 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 39086* Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝐹𝑉 → (𝐹X𝑥𝐴 𝐵𝐹:𝐴𝐵))

Theoremiineq1d 39087* Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 𝑥𝐴 𝐶 = 𝑥𝐵 𝐶)

Theoremmetpsmet 39088 A metric is a pseudometric. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))

Theoremixpssixp 39089 Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑X𝑥𝐴 𝐵X𝑥𝐴 𝐶)

Theoremballss3 39090* 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 39091* Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 𝑥𝐴 𝐵𝐶)

Theoremiunincfi 39092* 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 39093 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 39094 Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})

Theoremrexanuz3 39095* 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 39096* Equality theorem for restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴 = 𝐵)       (𝜑 → {𝑥𝐴𝜒} = {𝑥𝐵𝜒})

Theoremcbvmpt22 39097* 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 39098* 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)

Theoremeliuniin 39099* Indexed union of indexed intersections. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝐴 = 𝑥𝐵 𝑦𝐶 𝐷       (𝑍𝑉 → (𝑍𝐴 ↔ ∃𝑥𝐵𝑦𝐶 𝑍𝐷))

Theoremssabf 39100 Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
𝑥𝐴       (𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

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