Theorem colinrel 36090

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 6053	. 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 36072	. . 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 5718	. 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 2111  ∃wrex 3056  ⟨cop 4582   class class class wbr 5091  ^◡ccnv 5615  Rel wrel 5621  ‘cfv 6481  {coprab 7347  ℕcn 12122  𝔼cee 28864   Btwn cbtwn 28865   Colinear ccolin 36070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3919  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-colinear 36072

This theorem is referenced by:  brcolinear2  36091