P. 329 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isoun 32801*	Infer an isomorphism from a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)
⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥𝑅𝑦)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) → 𝑧𝑆𝑤)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑥𝑅𝑦)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐵) → ¬ 𝑧𝑆𝑤)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)    &   ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅)    ⇒   ⊢ (𝜑 → (𝐻 ∪ 𝐺) Isom 𝑅, 𝑆 ((𝐴 ∪ 𝐶), (𝐵 ∪ 𝐷)))
21.3.4.8  Disjointness (additional proof requiring functions)
Theorem	disjdsct 32802*	A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 6561) (Contributed by Thierry Arnoux, 28-Feb-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵))
21.3.4.9  First and second members of an ordered pair - misc additions
Theorem	df1stres 32803*	Definition for a restriction of the 1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥)
Theorem	df2ndres 32804*	Definition for a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦)
Theorem	1stpreimas 32805	The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (◡(1st ↾ 𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))
Theorem	1stpreima 32806	The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
⊢ (𝐴 ⊆ 𝐵 → (◡(1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))
Theorem	2ndpreima 32807	The preimage by 2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
⊢ (𝐴 ⊆ 𝐶 → (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))
Theorem	curry2ima 32808*	The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.)
⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶})))    ⇒   ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})
Theorem	preiman0 32809	The preimage of a nonempty set is nonempty. (Contributed by Thierry Arnoux, 9-Jun-2024.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ 𝐴) ≠ ∅)
Theorem	intimafv 32810*	The intersection of an image set, as an indexed intersection of function values. (Contributed by Thierry Arnoux, 15-Jun-2024.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ∩ (𝐹 “ 𝐴) = ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
21.3.4.10  Countable Sets
Theorem	snct 32811	A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≼ ω)
Theorem	prct 32812	An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ≼ ω)
Theorem	mpocti 32813*	An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.)
⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉    ⇒   ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ≼ ω)
Theorem	abrexct 32814*	An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≼ ω)
Theorem	mptctf 32815	A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω)
Theorem	abrexctf 32816*	An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≼ ω)
Theorem	padct 32817*	Index a countable set with integers and pad with 𝑍. (Contributed by Thierry Arnoux, 1-Jun-2020.) Avoid ax-rep 5206. (Revised by GG, 2-Apr-2026.)
⊢ ((𝐴 ≼ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
Theorem	f1od2 32818*	Sufficient condition for a binary function expressed in maps-to notation to be bijective. (Contributed by Thierry Arnoux, 17-Aug-2017.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌))    &   ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))    ⇒   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)
Theorem	fcobij 32819*	Composing functions with a bijection yields a bijection between sets of functions. (Contributed by Thierry Arnoux, 25-Aug-2017.)
⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑m 𝑅) ↦ (𝐺 ∘ 𝑓)):(𝑆 ↑m 𝑅)–1-1-onto→(𝑇 ↑m 𝑅))
Theorem	fcobijfs 32820*	Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 9318. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑆)    &   ⊢ 𝑄 = (𝐺‘𝑂)    &   ⊢ 𝑋 = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂}    &   ⊢ 𝑌 = {ℎ ∈ (𝑇 ↑m 𝑅) ∣ ℎ finSupp 𝑄}    ⇒   ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌)
Theorem	fcobijfs2 32821*	Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also fcobijfs 32820 and mapfien 9318. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐺:𝑅–1-1-onto→𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑇)    &   ⊢ 𝑋 = {𝑔 ∈ (𝑇 ↑m 𝑆) ∣ 𝑔 finSupp 𝑂}    &   ⊢ 𝑌 = {ℎ ∈ (𝑇 ↑m 𝑅) ∣ ℎ finSupp 𝑂}    ⇒   ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝑓 ∘ 𝐺)):𝑋–1-1-onto→𝑌)
Theorem	suppss3 32822*	Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑍) → 𝐵 = 𝑍)    ⇒   ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))
Theorem	fsuppcurry1 32823*	Finite support of a curried function with a constant first argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)
⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑍)    ⇒   ⊢ (𝜑 → 𝐺 finSupp 𝑍)
Theorem	fsuppcurry2 32824*	Finite support of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)
⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑍)    ⇒   ⊢ (𝜑 → 𝐺 finSupp 𝑍)
Theorem	offinsupp1 32825*	Finite support for a function operation. (Contributed by Thierry Arnoux, 8-Jul-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝑇)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑌𝑅𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) finSupp 𝑍)
Theorem	ffs2 32826	Rewrite a function's support based with its codomain rather than the universal class. See also fsuppeq 8122. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ 𝐶 = (𝐵 ∖ {𝑍})    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶))
Theorem	ffsrn 32827	The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)    ⇒   ⊢ (𝜑 → ran 𝐹 ∈ Fin)
Theorem	cocnvf1o 32828	Composing with the inverse of a bijection. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴–1-1-onto→𝐴)    ⇒   ⊢ (𝜑 → (𝐹 = (𝐺 ∘ 𝐻) ↔ 𝐺 = (𝐹 ∘ ◡𝐻)))
Theorem	resf1o 32829*	Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.)
⊢ 𝑋 = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}    &   ⊢ 𝐹 = (𝑓 ∈ 𝑋 ↦ (𝑓 ↾ 𝐶))    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝐹:𝑋–1-1-onto→(𝐵 ↑m 𝐶))
Theorem	maprnin 32830*	Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶}
Theorem	fpwrelmapffslem 32831*	Lemma for fpwrelmapffs 32833. For this theorem, the sets 𝐴 and 𝐵 could be infinite, but the relation 𝑅 itself is finite. (Contributed by Thierry Arnoux, 1-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝒫 𝐵)    &   ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})    ⇒   ⊢ (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
Theorem	fpwrelmap 32832*	Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 10368 and marypha2lem1 9345. (Contributed by Thierry Arnoux, 28-Aug-2017.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})    ⇒   ⊢ 𝑀:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
Theorem	fpwrelmapffs 32833*	Define a canonical mapping between finite relations (finite subsets of a cartesian product) and functions with finite support into finite subsets. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})    &   ⊢ 𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin}    ⇒   ⊢ (𝑀 ↾ 𝑆):𝑆–1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)
21.3.5  Real and Complex Numbers
Theorem	sgnval2 32834	Value of the signum of a real number, expresssed using absolute value. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (sgn‘𝐴) = (𝐴 / (abs‘𝐴)))
21.3.5.1  Complex operations - misc. additions
Theorem	creq0 32835	The real representation of complex numbers is zero iff both its terms are zero. Cf. crne0 12150. (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 = 0 ∧ 𝐵 = 0) ↔ (𝐴 + (i · 𝐵)) = 0))
Theorem	1nei 32836	The imaginary unit i is not one. (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ 1 ≠ i
Theorem	1neg1t1neg1 32837	An integer unit times itself. (Contributed by Thierry Arnoux, 23-Aug-2020.)
⊢ (𝑁 ∈ {-1, 1} → (𝑁 · 𝑁) = 1)
Theorem	nnmulge 32838	Multiplying by a positive integer 𝑀 yields greater than or equal nonnegative integers. (Contributed by Thierry Arnoux, 13-Dec-2021.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑀 · 𝑁))
Theorem	submuladdd 32839	The product of a difference and a sum. Cf. addmulsub 11610. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐴 · 𝐷)) − ((𝐵 · 𝐶) + (𝐵 · 𝐷))))
Theorem	binom2subadd 32840	The difference of the squares of the sum and difference of two complex numbers 𝐴 and 𝐵. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((𝐴 + 𝐵)↑2) − ((𝐴 − 𝐵)↑2)) = (4 · (𝐴 · 𝐵)))
Theorem	cjsubd 32841	Complex conjugate distributes over subtraction. (Contributed by Thierry Arnoux, 1-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵)))
Theorem	re0cj 32842	The conjugate of a pure imaginary number is its negative. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (ℜ‘𝐴) = 0)    ⇒   ⊢ (𝜑 → (∗‘𝐴) = -𝐴)
Theorem	receqid 32843	Real numbers equal to their own reciprocal have absolute value 1. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → ((1 / 𝐴) = 𝐴 ↔ (abs‘𝐴) = 1))
Theorem	pythagreim 32844	A simplified version of the Pythagorean theorem, where the points 𝐴 and 𝐵 respectively lie on the imaginary and real axes, and the right angle is at the origin. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((abs‘(𝐵 − (i · 𝐴)))↑2) = ((𝐴↑2) + (𝐵↑2)))
Theorem	efiargd 32845	The exponential of the "arg" function ℑ ∘ log, deduction version. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (exp‘(i · (ℑ‘(log‘𝐴)))) = (𝐴 / (abs‘𝐴)))
Theorem	arginv 32846	The argument of the inverse of a complex number 𝐴. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → ¬ -𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (ℑ‘(log‘(1 / 𝐴))) = -(ℑ‘(log‘𝐴)))
Theorem	argcj 32847	The argument of the conjugate of a complex number 𝐴. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → ¬ -𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (ℑ‘(log‘(∗‘𝐴))) = -(ℑ‘(log‘𝐴)))
Theorem	quad3d 32848	Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.) Deduction version. (Revised by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0)    ⇒   ⊢ (𝜑 → (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴))))
21.3.5.2  Ordering on reals - misc additions
Theorem	lt2addrd 32849*	If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐶))    ⇒   ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑐 ∈ ℝ (𝐴 = (𝑏 + 𝑐) ∧ 𝑏 < 𝐵 ∧ 𝑐 < 𝐶))
21.3.5.3  Extended reals - misc additions
Theorem	nn0mnfxrd 32850	Nonnegative integers or minus infinity are extended real numbers. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℕ0 ∪ {-∞}))    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ*)
Theorem	xrlelttric 32851	Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ∨ 𝐵 < 𝐴))
Theorem	xaddeq0 32852	Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 +𝑒 𝐵) = 0 ↔ 𝐴 = -𝑒𝐵))
Theorem	rexmul2 32853	If the result 𝐴 of an extended real multiplication is real, then its first factor 𝐵 is also real. See also rexmul 13221. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 0 < 𝐶)    &   ⊢ (𝜑 → 𝐴 = (𝐵 ·e 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 ∈ ℝ)
Theorem	xrinfm 32854	The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
⊢ inf(ℝ*, ℝ*, < ) = -∞
Theorem	le2halvesd 32855	A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ (𝐶 / 2))    &   ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 2))    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) ≤ 𝐶)
Theorem	xraddge02 32856	A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ 𝐵 → 𝐴 ≤ (𝐴 +𝑒 𝐵)))
Theorem	xrge0addge 32857	A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 19-Jul-2020.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 𝐴 ≤ (𝐴 +𝑒 𝐵))
Theorem	xlt2addrd 32858*	If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ≠ -∞)    &   ⊢ (𝜑 → 𝐶 ≠ -∞)    &   ⊢ (𝜑 → 𝐴 < (𝐵 +𝑒 𝐶))    ⇒   ⊢ (𝜑 → ∃𝑏 ∈ ℝ* ∃𝑐 ∈ ℝ* (𝐴 = (𝑏 +𝑒 𝑐) ∧ 𝑏 < 𝐵 ∧ 𝑐 < 𝐶))
Theorem	xrge0infss 32859*	Any subset of nonnegative extended reals has an infimum. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)
⊢ (𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
Theorem	xrge0infssd 32860	Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)
⊢ (𝜑 → 𝐶 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ (0[,]+∞))    ⇒   ⊢ (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < ))
Theorem	xrge0addcld 32861	Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019.)
⊢ (𝜑 → 𝐴 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞))
Theorem	xrge0subcld 32862	Condition for closure of nonnegative extended reals under subtraction. (Contributed by Thierry Arnoux, 27-May-2020.)
⊢ (𝜑 → 𝐴 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞))
Theorem	infxrge0lb 32863	A member of a set of nonnegative extended reals is greater than or equal to the set's infimum. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵)
Theorem	infxrge0glb 32864*	The infimum of a set of nonnegative extended reals is the greatest lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵))
Theorem	infxrge0gelb 32865*	The infimum of a set of nonnegative extended reals is greater than or equal to a lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
⊢ (𝜑 → 𝐴 ⊆ (0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))    ⇒   ⊢ (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥))
Theorem	xrofsup 32866	The supremum is preserved by extended addition set operation. (Provided minus infinity is not involved as it does not behave well with addition.) (Contributed by Thierry Arnoux, 20-Mar-2017.)
⊢ (𝜑 → 𝑋 ⊆ ℝ*)    &   ⊢ (𝜑 → 𝑌 ⊆ ℝ*)    &   ⊢ (𝜑 → sup(𝑋, ℝ*, < ) ≠ -∞)    &   ⊢ (𝜑 → sup(𝑌, ℝ*, < ) ≠ -∞)    &   ⊢ (𝜑 → 𝑍 = ( +𝑒 “ (𝑋 × 𝑌)))    ⇒   ⊢ (𝜑 → sup(𝑍, ℝ*, < ) = (sup(𝑋, ℝ*, < ) +𝑒 sup(𝑌, ℝ*, < )))
Theorem	supxrnemnf 32867	The supremum of a nonempty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.)
⊢ ((𝐴 ⊆ ℝ* ∧ 𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) ≠ -∞)
21.3.5.4  Extended nonnegative integers - misc additions
Theorem	xnn0gt0 32868	Nonzero extended nonnegative integers are strictly greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ ((𝑁 ∈ ℕ0* ∧ 𝑁 ≠ 0) → 0 < 𝑁)
Theorem	xnn01gt 32869	An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than 1. (Contributed by Thierry Arnoux, 21-Nov-2023.)
⊢ (𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ {0, 1} ↔ 1 < 𝑁))
Theorem	nn0xmulclb 32870	Finite multiplication in the extended nonnegative integers. (Contributed by Thierry Arnoux, 30-Jul-2023.)
⊢ (((𝐴 ∈ ℕ0* ∧ 𝐵 ∈ ℕ0*) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → ((𝐴 ·e 𝐵) ∈ ℕ0 ↔ (𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0)))
Theorem	xnn0nn0d 32871	Conditions for an extended nonnegative integer to be a nonnegative integer. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ0*)    &   ⊢ (𝜑 → 𝑁 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝑁 ∈ ℕ0)
Theorem	xnn0nnd 32872	Conditions for an extended nonnegative integer to be a positive integer. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ0*)    &   ⊢ (𝜑 → 𝑁 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝑁)    ⇒   ⊢ (𝜑 → 𝑁 ∈ ℕ)
21.3.5.5  Real number intervals - misc additions
Theorem	joiniooico 32873	Disjoint joining an open interval with a closed-below, open-above interval to form a closed-below, open-above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ ∧ ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)))
Theorem	ubico 32874	A right-open interval does not contain its right endpoint. (Contributed by Thierry Arnoux, 5-Apr-2017.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ*) → ¬ 𝐵 ∈ (𝐴[,)𝐵))
Theorem	xeqlelt 32875	Equality in terms of 'less than or equal to', 'less than'. (Contributed by Thierry Arnoux, 5-Jul-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ ¬ 𝐴 < 𝐵)))
Theorem	eliccelico 32876	Relate elementhood to a closed interval with elementhood to the same closed-below, open-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵)))
Theorem	elicoelioo 32877	Relate elementhood to a closed-below, open-above interval with elementhood to the same open interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵))))
Theorem	iocinioc2 32878	Intersection between two open-below, closed-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 7-Aug-2017.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = (𝐵(,]𝐶))
Theorem	xrdifh 32879	Class difference of a half-open interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.)
⊢ 𝐴 ∈ ℝ*    ⇒   ⊢ (ℝ* ∖ (𝐴[,]+∞)) = (-∞[,)𝐴)
Theorem	iocinif 32880	Relate intersection of two open-below, closed-above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → ((𝐴(,]𝐶) ∩ (𝐵(,]𝐶)) = if(𝐴 < 𝐵, (𝐵(,]𝐶), (𝐴(,]𝐶)))
Theorem	difioo 32881	The difference between two open intervals sharing the same lower bound. (Contributed by Thierry Arnoux, 26-Sep-2017.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶))
Theorem	difico 32882	The difference between two closed-below, open-above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 13-Oct-2017.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝐴[,)𝐶) ∖ (𝐵[,)𝐶)) = (𝐴[,)𝐵))
21.3.5.6  Finite intervals of integers - misc additions
Theorem	uzssico 32883	Upper integer sets are a subset of the corresponding closed-below, open-above intervals. (Contributed by Thierry Arnoux, 29-Dec-2021.)
⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ (𝑀[,)+∞))
Theorem	fz2ssnn0 32884	A finite set of sequential integers that is a subset of ℕ0. (Contributed by Thierry Arnoux, 8-Dec-2021.)
⊢ (𝑀 ∈ ℕ0 → (𝑀...𝑁) ⊆ ℕ0)
Theorem	nndiffz1 32885	Upper set of the positive integers. (Contributed by Thierry Arnoux, 22-Aug-2017.)
⊢ (𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ≥‘(𝑁 + 1)))
Theorem	ssnnssfz 32886*	For any finite subset of ℕ, find a superset in the form of a set of sequential integers. (Contributed by Thierry Arnoux, 13-Sep-2017.)
⊢ (𝐴 ∈ (𝒫 ℕ ∩ Fin) → ∃𝑛 ∈ ℕ 𝐴 ⊆ (1...𝑛))
Theorem	fzm1ne1 32887	Elementhood of an integer and its predecessor in finite intervals of integers. (Contributed by Thierry Arnoux, 1-Jan-2024.)
⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ≠ 𝑀) → (𝐾 − 1) ∈ (𝑀...(𝑁 − 1)))
Theorem	fzspl 32888	Split the last element of a finite set of sequential integers. More generic than fzsuc 13523. (Contributed by Thierry Arnoux, 7-Nov-2016.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) = ((𝑀...(𝑁 − 1)) ∪ {𝑁}))
Theorem	fzdif2 32889	Split the last element of a finite set of sequential integers. More generic than fzsuc 13523. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) ∖ {𝑁}) = (𝑀...(𝑁 − 1)))
Theorem	fzodif2 32890	Split the last element of a half-open range of sequential integers. (Contributed by Thierry Arnoux, 5-Dec-2021.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀..^(𝑁 + 1)) ∖ {𝑁}) = (𝑀..^𝑁))
Theorem	fzodif1 32891	Set difference of two half-open range of sequential integers sharing the same starting value. (Contributed by Thierry Arnoux, 2-Oct-2023.)
⊢ (𝐾 ∈ (𝑀...𝑁) → ((𝑀..^𝑁) ∖ (𝑀..^𝐾)) = (𝐾..^𝑁))
Theorem	fzsplit3 32892	Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)
⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑀...𝑁) = ((𝑀...(𝐾 − 1)) ∪ (𝐾...𝑁)))
Theorem	nn0diffz0 32893	Upper set of the nonnegative integers. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ (𝑁 ∈ ℕ0 → (ℕ0 ∖ (0...𝑁)) = (ℤ≥‘(𝑁 + 1)))
Theorem	bcm1n 32894	The proportion of one binomial coefficient to another with 𝑁 decreased by 1. (Contributed by Thierry Arnoux, 9-Nov-2016.)
⊢ ((𝐾 ∈ (0...(𝑁 − 1)) ∧ 𝑁 ∈ ℕ) → (((𝑁 − 1)C𝐾) / (𝑁C𝐾)) = ((𝑁 − 𝐾) / 𝑁))
21.3.5.7  Half-open integer ranges - misc additions
Theorem	iundisjfi 32895*	Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 25540. (Contributed by Thierry Arnoux, 15-Feb-2017.)
⊢ Ⅎ𝑛𝐵    &   ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    ⇒   ⊢ ∪ 𝑛 ∈ (1..^𝑁)𝐴 = ∪ 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)
Theorem	iundisj2fi 32896*	A disjoint union is disjoint, finite version. Cf. iundisj2 25541. (Contributed by Thierry Arnoux, 16-Feb-2017.)
⊢ Ⅎ𝑛𝐵    &   ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    ⇒   ⊢ Disj 𝑛 ∈ (1..^𝑁)(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)
Theorem	iundisjcnt 32897*	Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.)
⊢ Ⅎ𝑛𝐵    &   ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))    ⇒   ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
Theorem	iundisj2cnt 32898*	A countable disjoint union is disjoint. Cf. iundisj2 25541. (Contributed by Thierry Arnoux, 16-Feb-2017.)
⊢ Ⅎ𝑛𝐵    &   ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝑀)))    ⇒   ⊢ (𝜑 → Disj 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
Theorem	f1ocnt 32899*	Given a countable set 𝐴, number its elements by providing a one-to-one mapping either with ℕ or an integer range starting from 1. The domain of the function can then be used with iundisjcnt 32897 or iundisj2cnt 32898. (Contributed by Thierry Arnoux, 25-Jul-2020.)
⊢ (𝐴 ≼ ω → ∃𝑓(𝑓:dom 𝑓–1-1-onto→𝐴 ∧ (dom 𝑓 = ℕ ∨ dom 𝑓 = (1..^((♯‘𝐴) + 1)))))
Theorem	fz1nnct 32900	NN and integer ranges starting from 1 are countable. (Contributed by Thierry Arnoux, 25-Jul-2020.)
⊢ ((𝐴 = ℕ ∨ 𝐴 = (1..^𝑀)) → 𝐴 ≼ ω)