Theorem btwncolinear5 31350

Description: Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
btwncolinear5	⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐶 Colinear ⟨𝐴, 𝐵⟩))

Proof of Theorem btwncolinear5

Step	Hyp	Ref	Expression
1		btwncolinear1 31346	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐴 Colinear ⟨𝐵, 𝐶⟩))
2		colinearperm4 31342	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Colinear ⟨𝐵, 𝐶⟩ ↔ 𝐶 Colinear ⟨𝐴, 𝐵⟩))
3	1, 2	sylibd 228	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → 𝐶 Colinear ⟨𝐴, 𝐵⟩))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  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-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-iota 5768  df-fv 5812  df-oprab 6553  df-colinear 31316

This theorem is referenced by:  btwnconn1lem12  31375  lineunray  31424  lineelsb2  31425