Theorem outsideofcol 35105

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 35093	. 2 ⊢ (𝑃OutsideOf⟨𝑄, 𝑅⟩ ↔ (𝑃 Colinear ⟨𝑄, 𝑅⟩ ∧ ¬ 𝑃 Btwn ⟨𝑄, 𝑅⟩))
2	1	simplbi 499	1 ⊢ (𝑃OutsideOf⟨𝑄, 𝑅⟩ → 𝑃 Colinear ⟨𝑄, 𝑅⟩)

Syntax hints:  ¬ wn 3   → wi 4  ⟨cop 4635   class class class wbr 5149   Btwn cbtwn 28147   Colinear ccolin 35009  OutsideOfcoutsideof 35091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-br 5150  df-outsideof 35092

This theorem is referenced by: (None)