Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  colinrel Structured version   Visualization version   GIF version

Theorem colinrel 31334
 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.
StepHypRef Expression
1 relcnv 5422 . 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 31316 . . 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⟩))}
32releqi 5125 . 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⟩))})
41, 3mpbir 220 1 Rel Colinear
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   ∈ wcel 1977  ∃wrex 2897  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037  Rel wrel 5043  ‘cfv 5804  {coprab 6550  ℕcn 10897  𝔼cee 25568   Btwn cbtwn 25569   Colinear ccolin 31314 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-colinear 31316 This theorem is referenced by:  brcolinear2  31335
 Copyright terms: Public domain W3C validator