P. 462 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46101-46200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nregmodelaxext 46101*	The Axiom of Extensionality ax-ext 2732 is true in the permutation model defined from 𝐹. This theorem is an immediate consequence of the fact that ax-ext 2732 holds in all permutation models and is provided as an illustration. (Contributed by Eric Schmidt, 16-Nov-2025.)
⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)
21.46.10  Isomorphism of finite ordinals and non-negative integers
Theorem	hashnna 46102	The ♯ function on ω preserves addition. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (♯‘(𝐴 +o 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
Theorem	hashnnsuc 46103	The ♯ function on ω turns successor into adding 1. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ (𝐴 ∈ ω → (♯‘suc 𝐴) = ((♯‘𝐴) + 1))
Theorem	hashnnm 46104	The ♯ function on ω preserves multiplication. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (♯‘(𝐴 ·o 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))
Theorem	hashnnlt 46105	The ♯ function on ω preserves the ordering. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (♯‘𝐵) < (♯‘𝐴))
Theorem	hashnnltb 46106	The ♯ function on ω preserves the ordering. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ (♯‘𝐴) < (♯‘𝐵)))
Theorem	hashomf1o 46107	The ♯ function yields a bijection from ω to ℕ0. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ (♯ ↾ ω):ω–1-1-onto→ℕ0
Theorem	hashomiso 46108	The ♯ function yields an order isomorphism between ω and ℕ0. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ (♯ ↾ ω) Isom E , < (ω, ℕ0)
21.46.11  Hereditarily finite sets
Theorem	hfpr 46109	A pair whose members are hereditarily finite sets is a hereditarily finite set. (Contributed by Eric Schmidt, 26-Sep-2026.)
⊢ ((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → {𝐴, 𝐵} ∈ HF )
Theorem	hfxp 46110	The Cartesian product of two hereditarily finite sets is a hereditarily finite set. (Contributed by Eric Schmidt, 26-Sep-2026.)
⊢ ((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (𝐴 × 𝐵) ∈ HF )
Theorem	hfdm 46111	The domain of a hereditarily finite set is hereditarily finite. (Contributed by Eric Schmidt, 26-Sep-2026.)
⊢ (𝐴 ∈ HF → dom 𝐴 ∈ HF )
Theorem	hfrn 46112	The range of a hereditarily finite set is hereditarily finite. (Contributed by Eric Schmidt, 26-Sep-2026.)
⊢ (𝐴 ∈ HF → ran 𝐴 ∈ HF )
Theorem	hfrel 46113	A relation is a hereditarily finite set iff its domain and range are. (Contributed by Eric Schmidt, 26-Sep-2026.)
⊢ (Rel 𝑅 → (𝑅 ∈ HF ↔ (dom 𝑅 ∈ HF ∧ ran 𝑅 ∈ HF )))
Theorem	omhf 46114	Finite ordinals are hereditarily finite sets. (Contributed by Eric Schmidt, 26-Sep-2026.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ HF )
Theorem	omsshf 46115	The class of finite ordinals is included in the class of hereditarily finite sets. (Contributed by Eric Schmidt, 26-Sep-2026.)
⊢ ω ⊆ HF
21.46.12  Hereditarily finite structures
Syntax	chfstruct 46116	Extend class notation with the set of structures whose components are hereditarily finite.
class HFStruct
Definition	df-hfstruct 46117	Define the class of extensible structures whose components are all hereditarily finite. We will call these "HFStructs". Ideally, an HFStruct would itself be a hereditarily finite set, but this is not possible because the domain of a structure is, in the current formulation, a subset of ℕ. To get around this, we adjust the domain to be a subset of ω, obtaining the "converted form" (𝐹 ∘ (♯ ↾ ω)) of an HFStruct 𝐹. This is indeed hereditarily finite by hfstructhf 46123. Also, unlike general extensible structures, we do not allow ∅ to be a member of an HFStruct. This restriction gives us the property that two HFStructs are equal if and only if their converted forms are equal (hfstructcan 46124). (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
Theorem	hfstructstruct 46118*	An HFStruct is an extensible structure. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ (𝐹 ∈ HFStruct → ∃𝑥 𝐹 Struct 𝑥)
Theorem	hfstructfun 46119	An HFStruct is a function. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ (𝐹 ∈ HFStruct → Fun 𝐹)
Theorem	rnhfstructsshf 46120	The range of an HFStruct consists of hereditarily finite sets. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ (𝐹 ∈ HFStruct → ran 𝐹 ⊆ HF )
Theorem	ishfstruct 46121	Conditions that imply that 𝐹 is an HFStruct. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ ((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ HFStruct)
Theorem	rnhfstructhf 46122	The range of an HFStruct is a hereditarily finite set. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ (𝐹 ∈ HFStruct → ran 𝐹 ∈ HF )
Theorem	hfstructhf 46123	A converted HFStruct is a hereditarily finite set. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ (𝐹 ∈ HFStruct → (𝐹 ∘ (♯ ↾ ω)) ∈ HF )
Theorem	hfstructcan 46124	Two HFStructs are equal if and only if their converted forms are equal. (Contributed by Eric Schmidt, 29-Sep-2026.)
⊢ ((𝐹 ∈ HFStruct ∧ 𝐺 ∈ HFStruct) → ((𝐹 ∘ (♯ ↾ ω)) = (𝐺 ∘ (♯ ↾ ω)) ↔ 𝐹 = 𝐺))
21.47  Mathbox for Glauco Siliprandi
21.47.1  Miscellanea
Theorem	evth2f 46125*	A version of evth2 25286 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑦𝐹    &   ⊢ Ⅎ𝑥𝑋    &   ⊢ Ⅎ𝑦𝑋    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦))
Theorem	elunif 46126*	A version of eluni 4869 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
Theorem	rzalf 46127	A version of rzal 4449 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥 𝐴 = ∅    ⇒   ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	fvelrnbf 46128	A version of fvelrnb 6945 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐹    ⇒   ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
Theorem	rfcnpre1 46129	If F is a continuous function with respect to the standard topology, then the preimage A of the values greater 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 ∣ 𝐵 < (𝐹‘𝑥)}    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐽)
Theorem	ubelsupr 46130*	If U belongs to A and U is an upper bound, then U is the sup of A. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝑈 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ 𝑈) → 𝑈 = sup(𝐴, ℝ, < ))
Theorem	fsumcnf 46131*	A finite sum of functions to complex numbers from a common topological space is continuous, without disjoint var constraint x ph. The class expression for B normally contains free variables k and x to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (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	mulltgt0 46132	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)
Theorem	rspcegf 46133	A version of rspcev 3576 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)
Theorem	rabexgf 46134	A version of rabexg 5298 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	fcnre 46135	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)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
Theorem	sumsnd 46136*	A sum of a singleton is the term. The deduction version of sumsn 15917. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ (𝜑 → Ⅎ𝑘𝐵)    &   ⊢ Ⅎ𝑘𝜑    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)
Theorem	evthf 46137*	A version of evth 25285 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑦𝐹    &   ⊢ Ⅎ𝑥𝑋    &   ⊢ Ⅎ𝑦𝑋    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))
Theorem	cnfex 46138	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)
Theorem	fnchoice 46139*	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 46140*	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 25196 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 46141	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 46142*	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 46143*	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 46144*	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 46145*	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 46146*	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 46147*	This is the core Lemma for refsum2cn 46148: 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 46148*	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 46149	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 46150	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 46151	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 46152	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 46153	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 46154	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 46155	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 46156	Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (¬ 𝜑 → 𝜓)    &   ⊢ (¬ 𝜑 → ¬ 𝜓)    ⇒   ⊢ 𝜑
Theorem	iuneq2df 46157	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 46158*	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 46159	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 46160	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 46161*	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 46162	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 46163*	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 46164	The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ∪ {∅} = ∅
Theorem	ssin0 46165	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 46166	Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶))
Theorem	pwpwuni 46167	Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴 ∈ 𝒫 𝐵))
Theorem	disjiun2 46168*	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 46169	The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin)
Theorem	ssinss2d 46170	Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶)
Theorem	zct 46171	The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ℤ ≼ ω
Theorem	pwfin0 46172	A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅
Theorem	uzct 46173	An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ 𝑍 = (ℤ≥‘𝑁)    ⇒   ⊢ 𝑍 ≼ ω
Theorem	iunxsnf 46174*	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 46175*	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 46176*	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 46177*	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 46178	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 46179*	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 46180*	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 46181*	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 46182	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 46183*	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 46184*	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 46185*	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 46186	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 46187	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 46188*	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 46189*	Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵)
Theorem	ssd 46190*	A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	ralimralim 46191	Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))
Theorem	snelmap 46192	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 46193	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	iuneq1i 46194	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	ssinc 46195*	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 46196*	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 46197*	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 46198*	Equality theorem for indexed intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶)
Theorem	metpsmet 46199	A metric is a pseudometric. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (PsMet‘𝑋))
Theorem	ixpssixp 46200	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)