Theorem outsideofcol 36134

Description: Outside of implies colinearity. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
outsideofcol	⊢ (𝑃OutsideOf⟨𝑄, 𝑅⟩ → 𝑃 Colinear ⟨𝑄, 𝑅⟩)

Proof of Theorem outsideofcol

Step	Hyp	Ref	Expression
1		broutsideof 36122	. 2 ⊢ (𝑃OutsideOf⟨𝑄, 𝑅⟩ ↔ (𝑃 Colinear ⟨𝑄, 𝑅⟩ ∧ ¬ 𝑃 Btwn ⟨𝑄, 𝑅⟩))
2	1	simplbi 497	1 ⊢ (𝑃OutsideOf⟨𝑄, 𝑅⟩ → 𝑃 Colinear ⟨𝑄, 𝑅⟩)

Syntax hints:  ¬ wn 3   → wi 4  ⟨cop 4632   class class class wbr 5143   Btwn cbtwn 28904   Colinear ccolin 36038  OutsideOfcoutsideof 36120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-br 5144  df-outsideof 36121

This theorem is referenced by: (None)