Metamath Proof Explorer


Theorem btwncolinear5

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

Ref Expression
Assertion btwncolinear5 N A 𝔼 N B 𝔼 N C 𝔼 N C Btwn A B C Colinear A B

Proof

Step Hyp Ref Expression
1 btwncolinear1 N A 𝔼 N B 𝔼 N C 𝔼 N C Btwn A B A Colinear B C
2 colinearperm4 N A 𝔼 N B 𝔼 N C 𝔼 N A Colinear B C C Colinear A B
3 1 2 sylibd N A 𝔼 N B 𝔼 N C 𝔼 N C Btwn A B C Colinear A B