P. 364 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36301-36400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	btwnconn1lem12 36301	Lemma for btwnconn1 36304. Using a long string of invocations of linecgr 36284, we show that 𝐷 = 𝑑. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩))) ∧ ((𝐸 Btwn ⟨𝐶, 𝑐⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝑑⟩) ∧ ((𝐶 Btwn ⟨𝑐, 𝑃⟩ ∧ ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑑⟩) ∧ (𝐶 Btwn ⟨𝑑, 𝑅⟩ ∧ ⟨𝐶, 𝑅⟩Cgr⟨𝐶, 𝐸⟩) ∧ (𝑅 Btwn ⟨𝑃, 𝑄⟩ ∧ ⟨𝑅, 𝑄⟩Cgr⟨𝑅, 𝑃⟩))))) → 𝐷 = 𝑑)
Theorem	btwnconn1lem13 36302	Lemma for btwnconn1 36304. Begin back-filling and eliminating hypotheses. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ ((((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁)) ∧ (𝑑 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁)))) ∧ (((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩)) ∧ ((𝐷 Btwn ⟨𝐴, 𝑐⟩ ∧ ⟨𝐷, 𝑐⟩Cgr⟨𝐶, 𝐷⟩) ∧ (𝐶 Btwn ⟨𝐴, 𝑑⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝐶, 𝐷⟩)) ∧ ((𝑐 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑐, 𝑏⟩Cgr⟨𝐶, 𝐵⟩) ∧ (𝑑 Btwn ⟨𝐴, 𝑏⟩ ∧ ⟨𝑑, 𝑏⟩Cgr⟨𝐷, 𝐵⟩)))) → (𝐶 = 𝑐 ∨ 𝐷 = 𝑑))
Theorem	btwnconn1lem14 36303	Lemma for btwnconn1 36304. Final statement of the theorem when 𝐵 ≠ 𝐶. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩))) → (𝐶 Btwn ⟨𝐴, 𝐷⟩ ∨ 𝐷 Btwn ⟨𝐴, 𝐶⟩))
Theorem	btwnconn1 36304	Connectitivy law for betweenness. Theorem 5.1 of [Schwabhauser] p. 39-41. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝐵 ∧ 𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩) → (𝐶 Btwn ⟨𝐴, 𝐷⟩ ∨ 𝐷 Btwn ⟨𝐴, 𝐶⟩)))
Theorem	btwnconn2 36305	Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝐵 ∧ 𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐵 Btwn ⟨𝐴, 𝐷⟩) → (𝐶 Btwn ⟨𝐵, 𝐷⟩ ∨ 𝐷 Btwn ⟨𝐵, 𝐶⟩)))
Theorem	btwnconn3 36306	Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn ⟨𝐴, 𝐷⟩ ∧ 𝐶 Btwn ⟨𝐴, 𝐷⟩) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩)))
Theorem	midofsegid 36307	If two points fall in the same place in the middle of a segment, then they are identical. (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → ((𝐷 Btwn ⟨𝐴, 𝐵⟩ ∧ 𝐸 Btwn ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐷⟩Cgr⟨𝐴, 𝐸⟩) → 𝐷 = 𝐸))
Theorem	segcon2 36308*	Generalization of axsegcon 29015. This time, we generate an endpoint for a segment on the ray 𝑄𝐴 congruent to 𝐵𝐶 and starting at 𝑄, as opposed to axsegcon 29015, where the segment starts at 𝐴 (Contributed by Scott Fenton, 14-Oct-2013.) Remove unneeded inequality. (Revised by Scott Fenton, 15-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑄 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ∃𝑥 ∈ (𝔼‘𝑁)((𝐴 Btwn ⟨𝑄, 𝑥⟩ ∨ 𝑥 Btwn ⟨𝑄, 𝐴⟩) ∧ ⟨𝑄, 𝑥⟩Cgr⟨𝐵, 𝐶⟩))
21.11.17.6  Segment less than or equal to
Syntax	csegle 36309	Declare the constant for the segment less than or equal to relationship.
class Seg≤
Definition	df-segle 36310*	Define the segment length comparison relationship. This relationship expresses that the segment 𝐴𝐵 is no longer than 𝐶𝐷. In this section, we establish various properties of this relationship showing that it is a transitive, reflexive relationship on pairs of points that is substitutive under congruence. Definition 5.4 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ Seg≤ = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))}
Theorem	brsegle 36311*	Binary relation form of the segment comparison relationship. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝐶, 𝐷⟩ ∧ ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝑦⟩)))
Theorem	brsegle2 36312*	Alternate characterization of segment comparison. Theorem 5.5 of [Schwabhauser] p. 41-42. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ↔ ∃𝑥 ∈ (𝔼‘𝑁)(𝐵 Btwn ⟨𝐴, 𝑥⟩ ∧ ⟨𝐴, 𝑥⟩Cgr⟨𝐶, 𝐷⟩)))
Theorem	seglecgr12im 36313	Substitution law for segment comparison under congruence. Theorem 5.6 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩ ∧ ⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩) → ⟨𝐸, 𝐹⟩ Seg≤ ⟨𝐺, 𝐻⟩))
Theorem	seglecgr12 36314	Substitution law for segment comparison under congruence. Biconditional version. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩) → (⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ↔ ⟨𝐸, 𝐹⟩ Seg≤ ⟨𝐺, 𝐻⟩)))
Theorem	seglerflx 36315	Segment comparison is reflexive. Theorem 5.7 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐴, 𝐵⟩)
Theorem	seglemin 36316	Any segment is at least as long as a degenerate segment. Theorem 5.11 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐴, 𝐴⟩ Seg≤ ⟨𝐵, 𝐶⟩)
Theorem	segletr 36317	Segment less than is transitive. Theorem 5.8 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐸, 𝐹⟩) → ⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐸, 𝐹⟩))
Theorem	segleantisym 36318	Antisymmetry law for segment comparison. Theorem 5.9 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ∧ ⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐴, 𝐵⟩) → ⟨𝐴, 𝐵⟩Cgr⟨𝐶, 𝐷⟩))
Theorem	seglelin 36319	Linearity law for segment comparison. Theorem 5.10 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐶, 𝐷⟩ ∨ ⟨𝐶, 𝐷⟩ Seg≤ ⟨𝐴, 𝐵⟩))
Theorem	btwnsegle 36320	If 𝐵 falls between 𝐴 and 𝐶, then 𝐴𝐵 is no longer than 𝐴𝐶. (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ → ⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐴, 𝐶⟩))
Theorem	colinbtwnle 36321	Given three colinear points 𝐴, 𝐵, and 𝐶, 𝐵 falls in the middle iff the two segments to 𝐵 are no longer than 𝐴𝐶. Theorem 5.12 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ (⟨𝐴, 𝐵⟩ Seg≤ ⟨𝐴, 𝐶⟩ ∧ ⟨𝐵, 𝐶⟩ Seg≤ ⟨𝐴, 𝐶⟩))))
21.11.17.7  Outside-of relationship
Syntax	coutsideof 36322	Declare the syntax for the outside of constant.
class OutsideOf
Definition	df-outsideof 36323	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 )
Theorem	broutsideof 36324	Binary relation 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⟩))
Theorem	broutsideof2 36325	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⟩ ↔ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ (𝐴 Btwn ⟨𝑃, 𝐵⟩ ∨ 𝐵 Btwn ⟨𝑃, 𝐴⟩))))
Theorem	outsidene1 36326	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⟩ → 𝐴 ≠ 𝑃))
Theorem	outsidene2 36327	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⟩ → 𝐵 ≠ 𝑃))
Theorem	btwnoutside 36328	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))) → (((𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ 𝐶 ≠ 𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐶⟩) → (𝑃 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝑃OutsideOf⟨𝐴, 𝐵⟩)))
Theorem	broutsideof3 36329*	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⟩ ↔ (𝐴 ≠ 𝑃 ∧ 𝐵 ≠ 𝑃 ∧ ∃𝑐 ∈ (𝔼‘𝑁)(𝑐 ≠ 𝑃 ∧ 𝑃 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑃 Btwn ⟨𝐵, 𝑐⟩))))
Theorem	outsideofrflx 36330	Reflexivity 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)) → (𝐴 ≠ 𝑃 → 𝑃OutsideOf⟨𝐴, 𝐴⟩))
Theorem	outsideofcom 36331	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 36332	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 36333	Uniqueness law for OutsideOf. Analogue of segconeq 36213. (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 36334*	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 36335	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 36336	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 36337	Declare the constant for the line function.
class Line
Syntax	cray 36338	Declare the constant for the ray function.
class Ray
Syntax	clines2 36339	Declare the constant for the set of all lines.
class LinesEE
Definition	df-line2 36340*	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 36341*	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 36342	Define the set of all lines. Definition 6.14, part 2 of [Schwabhauser] p. 45. See ellines 36355 for membership. (Contributed by Scott Fenton, 28-Oct-2013.)
⊢ LinesEE = ran Line
Theorem	funray 36343	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 36344*	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 36345	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 36346	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 36347*	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 36348	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 36349*	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 36350	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 36351	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 36352	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 36353	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 36354	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 36355*	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 36356	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 36357*	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 36358*	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 36359*	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 36360*	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 36361	Declare the syntax for the forward difference operator.
class △
Definition	df-fwddif 36362*	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 36363	Declare the syntax for the nth forward difference operator.
class △n
Definition	df-fwddifn 36364*	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 36365	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 36366*	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 36367	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 36368*	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 36369	The rank of the union of two sets. Closed form of rankun 9769. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
Theorem	ranksng 36370	The rank of a singleton. Closed form of ranksn 9767. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ (𝐴 ∈ 𝑉 → (rank‘{𝐴}) = suc (rank‘𝐴))
Theorem	rankelg 36371	The membership relation is inherited by the rank function. Closed form of rankel 9752. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
Theorem	rankpwg 36372	The rank of a power set. Closed form of rankpw 9756. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
Theorem	rank0 36373	The rank of the empty set is ∅. (Contributed by Scott Fenton, 17-Jul-2015.)
⊢ (rank‘∅) = ∅
Theorem	rankeq1o 36374	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 36375	The constant Hf is a class.
class Hf
Definition	df-hf 36376	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 36377*	Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.)
⊢ (𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
Theorem	elhf2 36378	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 36379	Hereditarily finiteness via rank. Closed form of elhf2 36378. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))
Theorem	0hf 36380	The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
⊢ ∅ ∈ Hf
Theorem	hfun 36381	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 36382	The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ (𝐴 ∈ Hf → {𝐴} ∈ Hf )
Theorem	hfadj 36383	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 36384	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 36385	The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ Tr Hf
Theorem	hfext 36386*	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 36387	The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ (𝐴 ∈ Hf → ∪ 𝐴 ∈ Hf )
Theorem	hfpw 36388	The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ (𝐴 ∈ Hf → 𝒫 𝐴 ∈ Hf )
Theorem	hfninf 36389	ω is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ ¬ ω ∈ Hf
21.12  Mathbox for Gino Giotto
21.12.1  Equality theorems
21.12.1.1  Inference versions
Theorem	rmoeqi 36390	Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓)
Theorem	rmoeqbii 36391	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 36392	Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓)
Theorem	reueqbii 36393	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 36394	Formula-building inference for class substitution. General version of sbcbii 3786. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜓)
Theorem	disjeq1i 36395	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 36396	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 36397	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 36398	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 36399	Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶
Theorem	iineq12i 36400	Equality theorem for indexed intersection. Inference version. General version of iineq1i 36399. (Contributed by GG, 1-Sep-2025.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷