P. 367 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36601-36700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	outsideofcom 36601	Commutativity law for outsideness. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ 𝑃OutsideOf⟨𝐵, 𝐴⟩))
Theorem	outsideoftr 36602	Transitivity law for outsideness. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑃 ∈ (𝔼‘𝑁))) → ((𝑃OutsideOf⟨𝐴, 𝐵⟩ ∧ 𝑃OutsideOf⟨𝐵, 𝐶⟩) → 𝑃OutsideOf⟨𝐴, 𝐶⟩))
Theorem	outsideofeq 36603	Uniqueness law for OutsideOf. Analogue of segconeq 36483. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → (((𝐴OutsideOf⟨𝑋, 𝑅⟩ ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ (𝐴OutsideOf⟨𝑌, 𝑅⟩ ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) → 𝑋 = 𝑌))
Theorem	outsideofeu 36604*	Given a nondegenerate ray, there is a unique point congruent to the segment 𝐵𝐶 lying on the ray 𝐴𝑅. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 23-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ((𝑅 ≠ 𝐴 ∧ 𝐵 ≠ 𝐶) → ∃!𝑥 ∈ (𝔼‘𝑁)(𝐴OutsideOf⟨𝑥, 𝑅⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐵, 𝐶⟩)))
Theorem	outsidele 36605	Relate OutsideOf to Seg≤. Theorem 6.13 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ → (⟨𝑃, 𝐴⟩ Seg≤ ⟨𝑃, 𝐵⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩)))
Theorem	outsideofcol 36606	Outside of implies colinearity. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝑃OutsideOf⟨𝑄, 𝑅⟩ → 𝑃 Colinear ⟨𝑄, 𝑅⟩)
21.11.17.8  Lines and Rays
Syntax	cline2 36607	Declare the constant for the line function.
class Line
Syntax	cray 36608	Declare the constant for the ray function.
class Ray
Syntax	clines2 36609	Declare the constant for the set of all lines.
class LinesEE
Definition	df-line2 36610*	Define the Line function. This function generates the line passing through the distinct points 𝑎 and 𝑏. Adapted from definition 6.14 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 25-Oct-2013.)
⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
Definition	df-ray 36611*	Define the Ray function. This function generates the set of all points that lie on the ray starting at 𝑝 and passing through 𝑎. Definition 6.8 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 21-Oct-2013.)
⊢ Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
Definition	df-lines2 36612	Define the set of all lines. Definition 6.14, part 2 of [Schwabhauser] p. 45. See ellines 36625 for membership. (Contributed by Scott Fenton, 28-Oct-2013.)
⊢ LinesEE = ran Line
Theorem	funray 36613	Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Fun Ray
Theorem	fvray 36614*	Calculate the value of the Ray function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
Theorem	funline 36615	Show that the Line relationship is a function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ Fun Line
Theorem	linedegen 36616	When Line is applied with the same argument, the result is the empty set. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝐴Line𝐴) = ∅
Theorem	fvline 36617*	Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})
Theorem	liness 36618	A line is a subset of the space its two points lie in. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) ⊆ (𝔼‘𝑁))
Theorem	fvline2 36619*	Alternate definition of a line. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})
Theorem	lineunray 36620	A line is composed of a point and the two rays emerging from it. Theorem 6.15 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑃 ≠ 𝑅)) → (𝑃 Btwn ⟨𝑄, 𝑅⟩ → (𝑃Line𝑄) = (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅))))
Theorem	lineelsb2 36621	If 𝑆 lies on 𝑃𝑄, then 𝑃𝑄 = 𝑃𝑆. Theorem 6.16 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑆)) → (𝑆 ∈ (𝑃Line𝑄) → (𝑃Line𝑄) = (𝑃Line𝑆)))
Theorem	linerflx1 36622	Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → 𝑃 ∈ (𝑃Line𝑄))
Theorem	linecom 36623	Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → (𝑃Line𝑄) = (𝑄Line𝑃))
Theorem	linerflx2 36624	Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → 𝑄 ∈ (𝑃Line𝑄))
Theorem	ellines 36625*	Membership in the set of all lines. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝 ≠ 𝑞 ∧ 𝐴 = (𝑝Line𝑞)))
Theorem	linethru 36626	If 𝐴 is a line containing two distinct points 𝑃 and 𝑄, then 𝐴 is the line through 𝑃 and 𝑄. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ LinesEE ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐴 = (𝑃Line𝑄))
Theorem	hilbert1.1 36627*	There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → ∃𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))
Theorem	hilbert1.2 36628*	There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by NM, 17-Jun-2017.)
⊢ (𝑃 ≠ 𝑄 → ∃*𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))
Theorem	linethrueu 36629*	There is a unique line going through any two distinct points. Theorem 6.19 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝑄)) → ∃!𝑥 ∈ LinesEE (𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥))
Theorem	lineintmo 36630*	Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴 ≠ 𝐵) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
21.11.18  Forward difference
Syntax	cfwddif 36631	Declare the syntax for the forward difference operator.
class △
Definition	df-fwddif 36632*	Define the forward difference operator. This is a discrete analogue of the derivative operator. Definition 2.42 of [GramKnuthPat], p. 47. (Contributed by Scott Fenton, 18-May-2020.)
⊢ △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
Syntax	cfwddifn 36633	Declare the syntax for the nth forward difference operator.
class △n
Definition	df-fwddifn 36634*	Define the nth forward difference operator. This works out to be the forward difference operator iterated 𝑛 times. (Contributed by Scott Fenton, 28-May-2020.)
⊢ △n = (𝑛 ∈ ℕ0, 𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘))))))
Theorem	fwddifval 36635	Calculate the value of the forward difference operator at a point. (Contributed by Scott Fenton, 18-May-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑋 + 1) ∈ 𝐴)    ⇒   ⊢ (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))
Theorem	fwddifnval 36636*	The value of the forward difference operator at a point. (Contributed by Scott Fenton, 28-May-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝑁 △n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
Theorem	fwddifn0 36637	The value of the n-iterated forward difference operator at zero is just the function value. (Contributed by Scott Fenton, 28-May-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((0 △n 𝐹)‘𝑋) = (𝐹‘𝑋))
Theorem	fwddifnp1 36638*	The value of the n-iterated forward difference at a successor. (Contributed by Scott Fenton, 28-May-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑋 + 𝑘) ∈ 𝐴)    ⇒   ⊢ (𝜑 → (((𝑁 + 1) △n 𝐹)‘𝑋) = (((𝑁 △n 𝐹)‘(𝑋 + 1)) − ((𝑁 △n 𝐹)‘𝑋)))
21.11.19  Rank theorems
Theorem	rankung 36639	The rank of the union of two sets. Closed form of rankun 9829. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
Theorem	ranksng 36640	The rank of a singleton. Closed form of ranksn 9827. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ (𝐴 ∈ 𝑉 → (rank‘{𝐴}) = suc (rank‘𝐴))
Theorem	rankelg 36641	The membership relation is inherited by the rank function. Closed form of rankel 9812. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
Theorem	rankpwg 36642	The rank of a power set. Closed form of rankpw 9816. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
Theorem	rank0 36643	The rank of the empty set is ∅. (Contributed by Scott Fenton, 17-Jul-2015.)
⊢ (rank‘∅) = ∅
Theorem	rankeq1o 36644	The only set with rank 1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
⊢ ((rank‘𝐴) = 1o ↔ 𝐴 = {∅})
21.11.20  Hereditarily Finite Sets
Syntax	chf 36645	The constant Hf is a class.
class Hf
Definition	df-hf 36646	Define the hereditarily finite sets. These are the finite sets whose elements are finite, and so forth. (Contributed by Scott Fenton, 9-Jul-2015.)
⊢ Hf = ∪ (𝑅1 “ ω)
Theorem	elhf 36647*	Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.)
⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
Theorem	elhf2 36648	Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)
Theorem	elhf2g 36649	Hereditarily finiteness via rank. Closed form of elhf2 36648. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))
Theorem	0hf 36650	The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
⊢ ∅ ∈ Hf
Theorem	hfun 36651	The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ 𝐵) ∈ Hf )
Theorem	hfsn 36652	The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ (𝐴 ∈ Hf → {𝐴} ∈ Hf )
Theorem	hfadj 36653	Adjoining one HF element to an HF set preserves HF status. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ {𝐵}) ∈ Hf )
Theorem	hfelhf 36654	Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ Hf ) → 𝐴 ∈ Hf )
Theorem	hftr 36655	The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ Tr Hf
Theorem	hfext 36656*	Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ ((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ Hf (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)))
Theorem	hfuni 36657	The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ Hf )
Theorem	hfpw 36658	The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ (𝐴 ∈ Hf → 𝒫 𝐴 ∈ Hf )
Theorem	hfninf 36659	ω is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ ¬ ω ∈ Hf
21.11.21  Natural ordinal operations
Syntax	cnmul 36660	Declare the syntax for natural multiplication.
class ·no
Definition	df-nmul 36661*	Define natural ordinal multiplication. This is the corresponding operation to df-nadd 8653. (Contributed by Scott Fenton, 2-Jun-2026.)
⊢ ·no = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), (𝑝 ∈ V, 𝑚 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑎⦌⦋(2nd ‘𝑝) / 𝑏⦌∩ {𝑧 ∈ On ∣ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 ((𝑐𝑚𝑏) +no (𝑎𝑚𝑑)) ∈ (𝑧 +no (𝑐𝑚𝑑))}))
Theorem	nmulfn 36662	Natural multiplication is a function over pairs of ordinals. (Contributed by Scott Fenton, 2-Jun-2026.)
⊢ ·no Fn (On × On)
Theorem	nmulprop 36663*	Show closure and value of natural multiplication. (Contributed by Scott Fenton, 2-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·no 𝐵) ∈ On ∧ (𝐴 ·no 𝐵) = ∩ {𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·no 𝐵) +no (𝐴 ·no 𝑏)) ∈ (𝑥 +no (𝑎 ·no 𝑏))}))
Theorem	nmulcl 36664	Closure law for natural multiplication. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) ∈ On)
Theorem	nmulval 36665*	Show the value of natural multiplication. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) = ∩ {𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·no 𝐵) +no (𝐴 ·no 𝑏)) ∈ (𝑥 +no (𝑎 ·no 𝑏))})
Theorem	nmulcld 36666	Closure law for natural multiplication. Deduction form. (Contributed by Scott Fenton, 12-Jun-2026.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 ·no 𝐵) ∈ On)
Theorem	nmulcom 36667	Natural multiplication is commutative. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) = (𝐵 ·no 𝐴))
Theorem	nmulr0 36668	Natural multiplication by zero. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ (𝐴 ∈ On → (𝐴 ·no ∅) = ∅)
Theorem	nmull0 36669	Natural multiplication by zero. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ (𝐴 ∈ On → (∅ ·no 𝐴) = ∅)
Theorem	nmuladdel 36670	Ordering relationship for natural ordinal operations. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
Theorem	nmuladdss 36671	Ordering relationship for natural ordinal operations. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
Theorem	nmulss1 36672	Natural multiplication preserves less-than or equal. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ·no 𝐴) ⊆ (𝐶 ·no 𝐵))
Theorem	nmulel1 36673	Natural multiplication by a non-zero number preserves less-than. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ≠ ∅)) → (𝐶 ·no 𝐴) ∈ (𝐶 ·no 𝐵))
Theorem	ltnmul 36674*	Characterize less-than a natural product. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ (𝐵 ·no 𝐶) ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 (𝐴 +no (𝑏 ·no 𝑐)) ⊆ ((𝑏 ·no 𝐶) +no (𝐵 ·no 𝑐))))
Theorem	nmulle 36675*	A condition for bounding a natural product above. Converse of ltnmul 36674. (Contributed by Scott Fenton, 16-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·no 𝐵) ⊆ 𝐶 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·no 𝐵) +no (𝐴 ·no 𝑏)) ∈ (𝐶 +no (𝑎 ·no 𝑏))))
Theorem	ltnadd 36676*	Condition for bounding a natural sum below. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ (𝐵 +no 𝐶) ↔ (∃𝑏 ∈ 𝐵 𝐴 ⊆ (𝑏 +no 𝐶) ∨ ∃𝑐 ∈ 𝐶 𝐴 ⊆ (𝐵 +no 𝑐))))
Theorem	naddle 36677*	Condition for bounding natural addition above. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) ⊆ 𝐶 ↔ (∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶)))
Theorem	nmulrid 36678	Identity law for natural multiplication. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ (𝐴 ∈ On → (𝐴 ·no 1o) = 𝐴)
Theorem	nmullid 36679	Identity law for natural multiplication. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ (𝐴 ∈ On → (1o ·no 𝐴) = 𝐴)
21.12  Mathbox for Gino Giotto
21.12.1  Equality theorems
21.12.1.1  Inference versions
Theorem	rmoeqi 36680	Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓)
Theorem	rmoeqbii 36681	Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒)
Theorem	reueqi 36682	Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓)
Theorem	reueqbii 36683	Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒)
Theorem	sbceqbii 36684	Formula-building inference for class substitution. General version of sbcbii 3801. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜓)
Theorem	disjeq1i 36685	Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)
Theorem	disjeq12i 36686	Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)
Theorem	rabeqbii 36687	Equality theorem for restricted class abstractions. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓}
Theorem	iuneq12i 36688	Equality theorem for indexed union. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷
Theorem	iineq1i 36689	Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶
Theorem	iineq12i 36690	Equality theorem for indexed intersection. Inference version. General version of iineq1i 36689. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷
Theorem	riotaeqbii 36691	Equivalent wff's and equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓)
Theorem	riotaeqi 36692	Equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑)
Theorem	ixpeq1i 36693	Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶
Theorem	ixpeq12i 36694	Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷
Theorem	sumeq2si 36695	Equality inference for sum. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶
Theorem	sumeq12si 36696	Equality inference for sum. General version of sumeq2si 36695. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ Σ𝑥 ∈ 𝐴 𝐶 = Σ𝑥 ∈ 𝐵 𝐷
Theorem	prodeq2si 36697	Equality inference for product. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶
Theorem	prodeq12si 36698	Equality inference for product. General version of prodeq2si 36697. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷
Theorem	itgeq12i 36699	Equality inference for an integral. General version of itgeq1i 36700 and itgeq2i 36701. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥
Theorem	itgeq1i 36700	Equality inference for an integral. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥