Theorem outsideofcol 36646

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 36634	. 2 ⊢ (𝑃OutsideOf⟨𝑄, 𝑅⟩ ↔ (𝑃 Colinear ⟨𝑄, 𝑅⟩ ∧ ¬ 𝑃 Btwn ⟨𝑄, 𝑅⟩))
2	1	simplbi 502	1 ⊢ (𝑃OutsideOf⟨𝑄, 𝑅⟩ → 𝑃 Colinear ⟨𝑄, 𝑅⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ⟨cop 4600   class class class wbr 5114   Btwn cbtwn 29275   Colinear ccolin 36550  OutsideOfcoutsideof 36632

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-dif 3911  df-br 5115  df-outsideof 36633

This theorem is used by: (None)