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Theorem List for Metamath Proof Explorer - 32201-32300   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.8.32.7  Outside of relationship

Syntaxcoutsideof 32201 Declare the syntax for the outside of constant.
class OutsideOf

Definitiondf-outsideof 32202 The outside of relationship. This relationship expresses that 𝑃, 𝐴, and 𝐵 fall on a line, but 𝑃 is not on the segment 𝐴𝐵. This definition is taken from theorem 6.4 of [Schwabhauser] p. 43, since it requires no dummy variables. (Contributed by Scott Fenton, 17-Oct-2013.)
OutsideOf = ( Colinear ∖ Btwn )

Theorembroutsideof 32203 Binary relationship form of OutsideOf. Theorem 6.4 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐵⟩ ∧ ¬ 𝑃 Btwn ⟨𝐴, 𝐵⟩))

Theorembroutsideof2 32204 Alternate form of OutsideOf. Definition 6.1 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝑃OutsideOf⟨𝐴, 𝐵⟩ ↔ (𝐴𝑃𝐵𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))

Theoremoutsidene1 32205 Outsideness implies inequality. (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⟩ → 𝐴𝑃))

Theoremoutsidene2 32206 Outsideness implies inequality. (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⟩ → 𝐵𝑃))

Theorembtwnoutside 32207 A principle linking outsideness to betweenness. Theorem 6.2 of [Schwabhauser] p. 43. (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))) → (((𝐴𝑃𝐵𝑃𝐶𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐶⟩) → (𝑃 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝑃OutsideOf⟨𝐴, 𝐵⟩)))

Theorembroutsideof3 32208* Characterization of outsideness in terms of relationship to a fourth point. Theorem 6.3 of [Schwabhauser] p. 43. (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⟩ ↔ (𝐴𝑃𝐵𝑃 ∧ ∃𝑐 ∈ (𝔼‘𝑁)(𝑐𝑃𝑃 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑃 Btwn ⟨𝐵, 𝑐⟩))))

Theoremoutsideofrflx 32209 Reflexitivity of outsideness. Theorem 6.5 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)) → (𝐴𝑃𝑃OutsideOf⟨𝐴, 𝐴⟩))

Theoremoutsideofcom 32210 Commutitivity 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⟩))

Theoremoutsideoftr 32211 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⟩))

Theoremoutsideofeq 32212 Uniqueness law for OutsideOf. Analogue of segconeq 32092. (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))

Theoremoutsideofeu 32213* Given a non-degenerate 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))) → ((𝑅𝐴𝐵𝐶) → ∃!𝑥 ∈ (𝔼‘𝑁)(𝐴OutsideOf⟨𝑥, 𝑅⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐵, 𝐶⟩)))

Theoremoutsidele 32214 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⟩ Seg𝑃, 𝐵⟩ ↔ 𝐴 Btwn ⟨𝑃, 𝐵⟩)))

Theoremoutsideofcol 32215 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⟩)

20.8.32.8  Lines and Rays

Syntaxcline2 32216 Declare the constant for the line function.
class Line

Syntaxcray 32217 Declare the constant for the ray function.
class Ray

Syntaxclines2 32218 Declare the constant for the set of all lines.
class LinesEE

Definitiondf-line2 32219* 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) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}

Definitiondf-ray 32220* 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) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}

Definitiondf-lines2 32221 Define the set of all lines. Definition 6.14, part 2 of [Schwabhauser] p. 45. See ellines 32234 for membership. (Contributed by Scott Fenton, 28-Oct-2013.)
LinesEE = ran Line

Theoremfunray 32222 Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Ray

Theoremfvray 32223* 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) ∧ 𝑃𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})

Theoremfunline 32224 Show that the Line relationship is a function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Line

Theoremlinedegen 32225 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) = ∅

Theoremfvline 32226* 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) ∧ 𝐴𝐵)) → (𝐴Line𝐵) = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩})

Theoremliness 32227 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) ∧ 𝐴𝐵)) → (𝐴Line𝐵) ⊆ (𝔼‘𝑁))

Theoremfvline2 32228* 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) ∧ 𝐴𝐵)) → (𝐴Line𝐵) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})

Theoremlineunray 32229 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)) ∧ (𝑃𝑄𝑃𝑅)) → (𝑃 Btwn ⟨𝑄, 𝑅⟩ → (𝑃Line𝑄) = (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅))))

Theoremlineelsb2 32230 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) ∧ 𝑃𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃𝑆)) → (𝑆 ∈ (𝑃Line𝑄) → (𝑃Line𝑄) = (𝑃Line𝑆)))

Theoremlinerflx1 32231 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) ∧ 𝑃𝑄)) → 𝑃 ∈ (𝑃Line𝑄))

Theoremlinecom 32232 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) ∧ 𝑃𝑄)) → (𝑃Line𝑄) = (𝑄Line𝑃))

Theoremlinerflx2 32233 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) ∧ 𝑃𝑄)) → 𝑄 ∈ (𝑃Line𝑄))

Theoremellines 32234* 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)(𝑝𝑞𝐴 = (𝑝Line𝑞)))

Theoremlinethru 32235 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 ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝐴 = (𝑃Line𝑄))

Theoremhilbert1.1 32236* 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) ∧ 𝑃𝑄)) → ∃𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))

Theoremhilbert1.2 32237* 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.)
(𝑃𝑄 → ∃*𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))

Theoremlinethrueu 32238* 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) ∧ 𝑃𝑄)) → ∃!𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))

Theoremlineintmo 32239* 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 ∧ 𝐴𝐵) → ∃*𝑥(𝑥𝐴𝑥𝐵))

20.8.33  Forward difference

Syntaxcfwddif 32240 Declare the syntax for the forward difference operator.
class

Definitiondf-fwddif 32241* 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)) − (𝑓𝑥))))

Syntaxcfwddifn 32242 Declare the syntax for the nth forward difference operator.
class n

Definitiondf-fwddifn 32243* 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↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))))

Theoremfwddifval 32244 Calculate the value of the forward difference operator at a point. (Contributed by Scott Fenton, 18-May-2020.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋𝐴)    &   (𝜑 → (𝑋 + 1) ∈ 𝐴)       (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))

Theoremfwddifnval 32245* The value of the forward difference operator at a point. (Contributed by Scott Fenton, 28-May-2020.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴)       (𝜑 → ((𝑁n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))

Theoremfwddifn0 32246 The value of the n-iterated forward difference operator at zero is just the function value. (Contributed by Scott Fenton, 28-May-2020.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋𝐴)       (𝜑 → ((0 △n 𝐹)‘𝑋) = (𝐹𝑋))

Theoremfwddifnp1 32247* The value of the n-iterated forward difference at a successor. (Contributed by Scott Fenton, 28-May-2020.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   ((𝜑𝑘 ∈ (0...(𝑁 + 1))) → (𝑋 + 𝑘) ∈ 𝐴)       (𝜑 → (((𝑁 + 1) △n 𝐹)‘𝑋) = (((𝑁n 𝐹)‘(𝑋 + 1)) − ((𝑁n 𝐹)‘𝑋)))

20.8.34  Rank theorems

Theoremrankung 32248 The rank of the union of two sets. Closed form of rankun 8704. (Contributed by Scott Fenton, 15-Jul-2015.)
((𝐴𝑉𝐵𝑊) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Theoremranksng 32249 The rank of a singleton. Closed form of ranksn 8702. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴𝑉 → (rank‘{𝐴}) = suc (rank‘𝐴))

Theoremrankelg 32250 The membership relation is inherited by the rank function. Closed form of rankel 8687. (Contributed by Scott Fenton, 16-Jul-2015.)
((𝐵𝑉𝐴𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))

Theoremrankpwg 32251 The rank of a power set. Closed form of rankpw 8691. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

Theoremrank0 32252 The rank of the empty set is . (Contributed by Scott Fenton, 17-Jul-2015.)
(rank‘∅) = ∅

Theoremrankeq1o 32253 The only set with rank 1𝑜 is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
((rank‘𝐴) = 1𝑜𝐴 = {∅})

20.8.35  Hereditarily Finite Sets

Syntaxchf 32254 The constant Hf is a class.
class Hf

Definitiondf-hf 32255 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)

Theoremelhf 32256* Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.)
(𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))

Theoremelhf2 32257 Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
𝐴 ∈ V       (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)

Theoremelhf2g 32258 Hereditarily finiteness via rank. Closed form of elhf2 32257. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴𝑉 → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))

Theorem0hf 32259 The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
∅ ∈ Hf

Theoremhfun 32260 The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴𝐵) ∈ Hf )

Theoremhfsn 32261 The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴 ∈ Hf → {𝐴} ∈ Hf )

Theoremhfadj 32262 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 )

Theoremhfelhf 32263 Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.)
((𝐴𝐵𝐵 ∈ Hf ) → 𝐴 ∈ Hf )

Theoremhftr 32264 The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
Tr Hf

Theoremhfext 32265* 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 (𝑥𝐴𝑥𝐵)))

Theoremhfuni 32266 The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴 ∈ Hf → 𝐴 ∈ Hf )

Theoremhfpw 32267 The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴 ∈ Hf → 𝒫 𝐴 ∈ Hf )

Theoremhfninf 32268 ω is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
¬ ω ∈ Hf

20.9  Mathbox for Jeff Hankins

20.9.1  Miscellany

Theorema1i14 32269 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
(𝜓 → (𝜒𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theorema1i24 32270 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)
(𝜑 → (𝜒𝜏))       (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Theoremexp5d 32271 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(((𝜑𝜓) ∧ 𝜒) → ((𝜃𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp5g 32272 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑𝜓) → (((𝜒𝜃) ∧ 𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp5k 32273 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(𝜑 → (((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏) → 𝜂))       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp56 32274 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((((𝜑𝜓) ∧ 𝜒) ∧ (𝜃𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp58 32275 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(((𝜑𝜓) ∧ ((𝜒𝜃) ∧ 𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp510 32276 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑 ∧ (((𝜓𝜒) ∧ 𝜃) ∧ 𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp511 32277 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑 ∧ ((𝜓 ∧ (𝜒𝜃)) ∧ 𝜏)) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theoremexp512 32278 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((𝜑 ∧ ((𝜓𝜒) ∧ (𝜃𝜏))) → 𝜂)       (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))

Theorem3com12d 32279 Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑 → (𝜒𝜓𝜃))

Theoremimp5p 32280 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       (𝜑 → (𝜓 → ((𝜒𝜃𝜏) → 𝜂)))

Theoremimp5q 32281 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂)))))       ((𝜑𝜓) → ((𝜒𝜃𝜏) → 𝜂))

Theoremecase13d 32282 Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
(𝜑 → ¬ 𝜒)    &   (𝜑 → ¬ 𝜃)    &   (𝜑 → (𝜒𝜓𝜃))       (𝜑𝜓)

Theoremsubtr 32283 Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑌    &   𝑥𝑍    &   (𝑥 = 𝐴𝑋 = 𝑌)    &   (𝑥 = 𝐵𝑋 = 𝑍)       ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵𝑌 = 𝑍))

Theoremsubtr2 32284 Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝜓    &   𝑥𝜒    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       ((𝐴𝐶𝐵𝐷) → (𝐴 = 𝐵 → (𝜓𝜒)))

Theoremtrer 32285* A relation intersected with its converse is an equivalence relation if the relation is transitive. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
(∀𝑎𝑏𝑐((𝑎 𝑏𝑏 𝑐) → 𝑎 𝑐) → ( ) Er dom ( ))

Theoremelicc3 32286 An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ*𝐴𝐵 ∧ (𝐶 = 𝐴 ∨ (𝐴 < 𝐶𝐶 < 𝐵) ∨ 𝐶 = 𝐵))))

Theoremfinminlem 32287* A useful lemma about finite sets. If a property holds for a finite set, it holds for a minimal set. (Contributed by Jeff Hankins, 4-Dec-2009.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥 ∈ Fin 𝜑 → ∃𝑥(𝜑 ∧ ∀𝑦((𝑦𝑥𝜓) → 𝑥 = 𝑦)))

Theoremgtinf 32288* Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.) (Revised by AV, 10-Oct-2021.)
(((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑥𝑦) ∧ (𝐴 ∈ ℝ ∧ inf(𝑆, ℝ, < ) < 𝐴)) → ∃𝑧𝑆 𝑧 < 𝐴)

TheoremgtinfOLD 32289* Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.) Obsolete version of gtinf 32288 as of 10-Oct-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑆 𝑥𝑦) ∧ (𝐴 ∈ ℝ ∧ sup(𝑆, ℝ, < ) < 𝐴)) → ∃𝑧𝑆 𝑧 < 𝐴)

Theoremopnrebl 32290* A set is open in the standard topology of the reals precisely when every point can be enclosed in an open ball. (Contributed by Jeff Hankins, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
(𝐴 ∈ (topGen‘ran (,)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥𝐴𝑦 ∈ ℝ+ ((𝑥𝑦)(,)(𝑥 + 𝑦)) ⊆ 𝐴))

Theoremopnrebl2 32291* A set is open in the standard topology of the reals precisely when every point can be enclosed in an arbitrarily small ball. (Contributed by Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
(𝐴 ∈ (topGen‘ran (,)) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝑧𝑦 ∧ ((𝑥𝑧)(,)(𝑥 + 𝑧)) ⊆ 𝐴)))

Theoremnn0prpwlem 32292* Lemma for nn0prpw 32293. Use strong induction to show that every positive integer has unique prime power divisors. (Contributed by Jeff Hankins, 28-Sep-2013.)
(𝐴 ∈ ℕ → ∀𝑘 ∈ ℕ (𝑘 < 𝐴 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝𝑛) ∥ 𝑘 ↔ (𝑝𝑛) ∥ 𝐴)))

Theoremnn0prpw 32293* Two nonnegative integers are the same if and only if they are divisible by the same prime powers. (Contributed by Jeff Hankins, 29-Sep-2013.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝𝑛) ∥ 𝐴 ↔ (𝑝𝑛) ∥ 𝐵)))

20.9.2  Basic topological facts

Theoremtopbnd 32294 Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) = (((cls‘𝐽)‘𝐴) ∖ ((int‘𝐽)‘𝐴)))

Theoremopnbnd 32295 A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴𝐽 ↔ (𝐴 ∩ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴)))) = ∅))

Theoremcldbnd 32296 A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (((cls‘𝐽)‘𝐴) ∩ ((cls‘𝐽)‘(𝑋𝐴))) ⊆ 𝐴))

Theoremntruni 32297* A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋) → 𝑜𝑂 ((int‘𝐽)‘𝑜) ⊆ ((int‘𝐽)‘ 𝑂))

Theoremclsun 32298 A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝑋) → ((cls‘𝐽)‘(𝐴𝐵)) = (((cls‘𝐽)‘𝐴) ∪ ((cls‘𝐽)‘𝐵)))

Theoremclsint2 32299* The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝒫 𝑋) → ((cls‘𝐽)‘ 𝐶) ⊆ 𝑐𝐶 ((cls‘𝐽)‘𝑐))

Theoremopnregcld 32300* A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋) → (((cls‘𝐽)‘((int‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑜𝐽 𝐴 = ((cls‘𝐽)‘𝑜)))

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