Theorem btwncolinear1 36134

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 1333	. 2 ⊢ (𝐶 Btwn ⟨𝐴, 𝐵⟩ → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∨ 𝐵 Btwn ⟨𝐶, 𝐴⟩ ∨ 𝐶 Btwn ⟨𝐴, 𝐵⟩))
2		brcolinear 36124	. 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 246	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 395   ∨ w3o 1085   ∧ w3a 1086   ∈ wcel 2113  ⟨cop 4581   class class class wbr 5093  ‘cfv 6486  ℕcn 12132  𝔼cee 28867   Btwn cbtwn 28868   Colinear ccolin 36102

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 2115  ax-9 2123  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627  df-iota 6442  df-fv 6494  df-oprab 7356  df-colinear 36104

This theorem is referenced by:  btwncolinear2  36135  btwncolinear3  36136  btwncolinear4  36137  btwncolinear5  36138  btwncolinear6  36139  idinside  36149  btwnconn1lem12  36163  brsegle2  36174  broutsideof2  36187  outsidele  36197