P. 451 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45001-45100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fnchoice 45001*	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 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)))
Theorem	refsumcn 45002*	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 24810 for the analogous theorem on continuous complex functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
Theorem	rfcnpre2 45003	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.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵}    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐽)
Theorem	cncmpmax 45004*	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 (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑇 ≠ ∅)    ⇒   ⊢ (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < )))
Theorem	rfcnpre3 45005*	If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑡𝐹    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝑇 = ∪ 𝐽    &   ⊢ 𝐴 = {𝑡 ∈ 𝑇 ∣ 𝐵 ≤ (𝐹‘𝑡)}    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
Theorem	rfcnpre4 45006*	If F is a continuous function with respect to the standard topology, then the preimage A of the values less than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑡𝐹    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝑇 = ∪ 𝐽    &   ⊢ 𝐴 = {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ 𝐵}    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
Theorem	sumpair 45007*	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.)
⊢ (𝜑 → Ⅎ𝑘𝐷)    &   ⊢ (𝜑 → Ⅎ𝑘𝐸)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → 𝐸 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
Theorem	rfcnnnub 45008*	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.)
⊢ Ⅎ𝑡𝐹    &   ⊢ Ⅎ𝑡𝜑    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ 𝑇 = ∪ 𝐽    &   ⊢ (𝜑 → 𝑇 ≠ ∅)    &   ⊢ 𝐶 = (𝐽 Cn 𝐾)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑛)
Theorem	refsum2cnlem1 45009*	This is the core Lemma for refsum2cn 45010: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐺    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐴 = (𝑘 ∈ {1, 2} ↦ if(𝑘 = 1, 𝐹, 𝐺))    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ (𝐽 Cn 𝐾))
Theorem	refsum2cn 45010*	The sum of two continuus real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐺    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ (𝐽 Cn 𝐾))
Theorem	adantlllr 45011	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((((𝜑 ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	3adantlr3 45012	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜂)) ∧ 𝜃) → 𝜏)
Theorem	3adantll2 45013	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((((𝜑 ∧ 𝜂 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	3adantll3 45014	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	ssnel 45015	If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐴)
Theorem	sncldre 45016	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 (,))))
Theorem	n0p 45017	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)
Theorem	pm2.65ni 45018	Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (¬ 𝜑 → 𝜓)    &   ⊢ (¬ 𝜑 → ¬ 𝜓)    ⇒   ⊢ 𝜑
Theorem	iuneq2df 45019	Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶)
Theorem	nnfoctb 45020*	There exists a mapping from ℕ onto any (nonempty) countable set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝐴)
Theorem	elpwinss 45021	An element of the powerset of 𝐵 intersected with anything, is a subset of 𝐵. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐴 ∈ (𝒫 𝐵 ∩ 𝐶) → 𝐴 ⊆ 𝐵)
Theorem	unidmex 45022	If 𝐹 is a set, then ∪ dom 𝐹 is a set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ 𝑋 = ∪ dom 𝐹    ⇒   ⊢ (𝜑 → 𝑋 ∈ V)
Theorem	ndisj2 45023*	A non-disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅))
Theorem	zenom 45024	The set of integer numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ℤ ≈ ω
Theorem	uzwo4 45025*	Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑗𝜓    &   ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓)))
Theorem	unisn0 45026	The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ∪ {∅} = ∅
Theorem	ssin0 45027	If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) = ∅)
Theorem	inabs3 45028	Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶))
Theorem	pwpwuni 45029	Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))
Theorem	disjiun2 45030*	In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))    &   ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐸)    ⇒   ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅)
Theorem	0pwfi 45031	The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
Theorem	ssinss2d 45032	Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
Theorem	zct 45033	The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ℤ ≼ ω
Theorem	pwfin0 45034	A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅
Theorem	uzct 45035	An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝑍 = (ℤ≥‘𝑁)    ⇒   ⊢ 𝑍 ≼ ω
Theorem	iunxsnf 45036*	A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑥𝐶    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶
Theorem	fiiuncl 45037*	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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (𝑦 ∪ 𝑧) ∈ 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐷)
Theorem	iunp1 45038*	The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝐵    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∪ 𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (∪ 𝑘 ∈ (𝑀...𝑁)𝐴 ∪ 𝐵))
Theorem	fiunicl 45039*	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.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∪ 𝑦) ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴)
Theorem	ixpeq2d 45040	Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)
Theorem	disjxp1 45041*	The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 (𝐵 × 𝐶))
Theorem	disjsnxp 45042*	The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ Disj 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
Theorem	eliind 45043*	Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐵)    &   ⊢ (𝑥 = 𝐾 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐷)
Theorem	rspcef 45044	Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)
Theorem	ixpssmapc 45045*	An infinite Cartesian product is a subset of set exponentiation. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐶 ↑m 𝐴))
Theorem	elintd 45046*	Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)
Theorem	ssdf 45047*	A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	brneqtrd 45048	Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → ¬ 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴𝑅𝐶)
Theorem	ssnct 45049	A set containing an uncountable set is itself uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → ¬ 𝐴 ≼ ω)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐵 ≼ ω)
Theorem	ssuniint 45050*	Sufficient condition for being a subclass of the union of an intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ ∪ ∩ 𝐵)
Theorem	elintdv 45051*	Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)
Theorem	ssd 45052*	A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	ralimralim 45053	Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))
Theorem	snelmap 45054	Membership of the element in the range of a constant map. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → (𝐴 × {𝑥}) ∈ (𝐵 ↑m 𝐴))    ⇒   ⊢ (𝜑 → 𝑥 ∈ 𝐵)
Theorem	xrnmnfpnf 45055	An extended real that is neither real nor minus infinity, is plus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ -∞)    ⇒   ⊢ (𝜑 → 𝐴 = +∞)
Theorem	nelrnmpt 45056*	Non-membership in the range of a function in maps-to notaion. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ ran 𝐹)
Theorem	iuneq1i 45057	Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶
Theorem	nssrex 45058*	Negation of subclass relationship. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)
Theorem	ssinc 45059*	Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀..^𝑁)) → (𝐹‘𝑚) ⊆ (𝐹‘(𝑚 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑀) ⊆ (𝐹‘𝑁))
Theorem	ssdec 45060*	Inclusion relation for a monotonic sequence of sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑚 + 1)) ⊆ (𝐹‘𝑚))    ⇒   ⊢ (𝜑 → (𝐹‘𝑁) ⊆ (𝐹‘𝑀))
Theorem	elixpconstg 45061*	Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵))
Theorem	iineq1d 45062*	Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)
Theorem	metpsmet 45063	A metric is a pseudometric. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))
Theorem	ixpssixp 45064	Subclass theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)
Theorem	ballss3 45065*	A sufficient condition for a ball being a subset. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑋))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ*)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ (𝑃𝐷𝑥) < 𝑅) → 𝑥 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ 𝐴)
Theorem	iunincfi 45066*	Given a sequence of increasing sets, the union of a finite subsequence, is its last element. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑛 + 1)))    ⇒   ⊢ (𝜑 → ∪ 𝑛 ∈ (𝑀...𝑁)(𝐹‘𝑛) = (𝐹‘𝑁))
Theorem	nsstr 45067	If it's not a subclass, it's not a subclass of a smaller one. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ ((¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐶)
Theorem	rexanuz3 45068*	Combine two different upper integer properties into one, for a single integer. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑗𝜑    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜒)    &   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜓)    &   ⊢ (𝑘 = 𝑗 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑘 = 𝑗 → (𝜓 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝜃 ∧ 𝜏))
Theorem	cbvmpo2 45069*	Rule to change the second bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑤𝐴    &   ⊢ Ⅎ𝑤𝐶    &   ⊢ Ⅎ𝑦𝐸    &   ⊢ (𝑦 = 𝑤 → 𝐶 = 𝐸)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸)
Theorem	cbvmpo1 45070*	Rule to change the first bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑧𝐵    &   ⊢ Ⅎ𝑧𝐶    &   ⊢ Ⅎ𝑥𝐸    &   ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸)
Theorem	eliuniin 45071*	Indexed union of indexed intersections. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷    ⇒   ⊢ (𝑍 ∈ 𝑉 → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
Theorem	ssabf 45072	Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ⊆ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Theorem	pssnssi 45073	A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 ⊊ 𝐵    ⇒   ⊢ ¬ 𝐵 ⊆ 𝐴
Theorem	rabidim2 45074	Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑)
Theorem	eluni2f 45075*	Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)
Theorem	eliin2f 45076*	Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
Theorem	nssd 45077	Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ⊆ 𝐵)
Theorem	iineq12dv 45078*	Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)
Theorem	supxrcld 45079	The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ*)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) ∈ ℝ*)
Theorem	elrestd 45080	A sufficient condition for being an open set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐽 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐽)    &   ⊢ 𝐴 = (𝑋 ∩ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐵))
Theorem	eliuniincex 45081*	Counterexample to show that the additional conditions in eliuniin 45071 and eliuniin2 45092 are actually needed. Notice that the definition of 𝐴 is not even needed (it can be any class). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐵 = {∅}    &   ⊢ 𝐶 = ∅    &   ⊢ 𝐷 = ∅    &   ⊢ 𝑍 = V    ⇒   ⊢ ¬ (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷)
Theorem	eliincex 45082*	Counterexample to show that the additional conditions in eliin 4972 and eliin2 45088 are actually needed. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝐴 = V    &   ⊢ 𝐵 = ∅    ⇒   ⊢ ¬ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
Theorem	eliinid 45083*	Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)
Theorem	abssf 45084	Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴))
Theorem	supxrubd 45085	A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ 𝑆 = sup(𝐴, ℝ*, < )    ⇒   ⊢ (𝜑 → 𝐵 ≤ 𝑆)
Theorem	ssrabf 45086	Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑))
Theorem	ssrabdf 45087	Subclass of a restricted class abstraction (deduction form). (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})
Theorem	eliin2 45088*	Membership in indexed intersection. See eliincex 45082 for a counterexample showing that the precondition 𝐵 ≠ ∅ cannot be simply dropped. eliin 4972 uses an alternative precondition (and it doesn't have a disjoint var constraint between 𝐵 and 𝑥; see eliin2f 45076). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
Theorem	ssrab2f 45089	Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
Theorem	restuni3 45090	The underlying set of a subspace induced by the subspace operator ↾t. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))
Theorem	rabssf 45091	Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵))
Theorem	eliuniin2 45092*	Indexed union of indexed intersections. See eliincex 45082 for a counterexample showing that the precondition 𝐶 ≠ ∅ cannot be simply dropped. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Ⅎ𝑥𝐶    &   ⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷    ⇒   ⊢ (𝐶 ≠ ∅ → (𝑍 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 𝑍 ∈ 𝐷))
Theorem	restuni4 45093	The underlying set of a subspace induced by the ↾t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴)    ⇒   ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵)
Theorem	restuni6 45094	The underlying set of a subspace topology. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))
Theorem	restuni5 45095	The underlying set of a subspace induced by the ↾t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋) → 𝐴 = ∪ (𝐽 ↾t 𝐴))
Theorem	unirestss 45096	The union of an elementwise intersection is a subset of the underlying set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) ⊆ ∪ 𝐴)
Theorem	iniin1 45097*	Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ≠ ∅ → (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵))
Theorem	iniin2 45098*	Indexed intersection of intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝐴 ≠ ∅ → (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶))
Theorem	cbvrabv2 45099*	A more general version of cbvrabv 3426. Usage of this theorem is discouraged because it depends on ax-13 2376. Use of cbvrabv2w 45100 is preferred. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
Theorem	cbvrabv2w 45100*	A more general version of cbvrabv 3426. Version of cbvrabv2 45099 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Revised by GG, 14-Aug-2025.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}