Metamath Proof Explorer

Theorem btwncolinear4

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

Ref Expression
Assertion btwncolinear4 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → C Btwn A B → B Colinear C A

Proof

Step Hyp Ref Expression
1 btwncolinear1 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → C Btwn A B → A Colinear B C
2 colinearperm3 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Colinear B C ↔ B Colinear C A
3 1 2 sylibd ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → C Btwn A B → B Colinear C A