Theorem btwncolinear1 35718

Description: Betweenness implies colinearity. (Contributed by Scott Fenton, 7-Oct-2013.)

Assertion

Ref	Expression
btwncolinear1	⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐴 Colinear ⟨𝐵, 𝐶⟩))

Proof of Theorem btwncolinear1

Step	Hyp	Ref	Expression
1		3mix3 1329	. 2 ⊢ (𝐶 Btwn ⟨𝐴, 𝐵⟩ → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))
2		brcolinear 35708	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩)))
3	1, 2	imbitrrid 245	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐴 Colinear ⟨𝐵, 𝐶⟩))

Syntax hints:   → wi 4   ∧ wa 394   ∨ w3o 1083   ∧ w3a 1084   ∈ wcel 2098  ⟨cop 4631   class class class wbr 5144  ‘cfv 6543  ℕcn 12237  𝔼cee 28738   Btwn cbtwn 28739   Colinear ccolin 35686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-xp 5679  df-rel 5680  df-cnv 5681  df-iota 6495  df-fv 6551  df-oprab 7417  df-colinear 35688

This theorem is referenced by:  btwncolinear2  35719  btwncolinear3  35720  btwncolinear4  35721  btwncolinear5  35722  btwncolinear6  35723  idinside  35733  btwnconn1lem12  35747  brsegle2  35758  broutsideof2  35771  outsidele  35781