Theorem colinrel 36039

Description: Colinearity is a relationship. (Contributed by Scott Fenton, 7-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
colinrel	⊢ Rel Colinear

Proof of Theorem colinrel

Dummy variables 𝑞 𝑝 𝑟 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 6125	. 2 ⊢ Rel ◡{⟨⟨𝑞, 𝑟⟩, 𝑝⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑞, 𝑟⟩ ∨ 𝑞 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑞⟩))}
2		df-colinear 36021	. . 3 ⊢ Colinear = ◡{⟨⟨𝑞, 𝑟⟩, 𝑝⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑞, 𝑟⟩ ∨ 𝑞 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑞⟩))}
3	2	releqi 5790	. 2 ⊢ (Rel Colinear ↔ Rel ◡{⟨⟨𝑞, 𝑟⟩, 𝑝⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑞 ∈ (𝔼‘𝑛) ∧ 𝑟 ∈ (𝔼‘𝑛)) ∧ (𝑝 Btwn ⟨𝑞, 𝑟⟩ ∨ 𝑞 Btwn ⟨𝑟, 𝑝⟩ ∨ 𝑟 Btwn ⟨𝑝, 𝑞⟩))})
4	1, 3	mpbir 231	1 ⊢ Rel Colinear

Syntax hints:   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086   ∈ wcel 2106  ∃wrex 3068  ⟨cop 4637   class class class wbr 5148  ^◡ccnv 5688  Rel wrel 5694  ‘cfv 6563  {coprab 7432  ℕcn 12264  𝔼cee 28918   Btwn cbtwn 28919   Colinear ccolin 36019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-colinear 36021

This theorem is referenced by:  brcolinear2  36040