Theorem btwntriv1

Description: Betweenness always holds for the first endpoint. Theorem 3.3 of Schwabhauser p. 30. (Contributed by Scott Fenton, 12-Jun-2013)

	Ref	Expression
Assertion	btwntriv1	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐴 Btwn ⟨ 𝐴 , 𝐵 ⟩ )

Proof

Step	Hyp	Ref	Expression
1		btwntriv2	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐴 Btwn ⟨ 𝐵 , 𝐴 ⟩ )
2	1	3com23	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐴 Btwn ⟨ 𝐵 , 𝐴 ⟩ )
3		simp1	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝑁 ∈ ℕ )
4		simp2	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
5		simp3	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
6		btwncom	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐴 Btwn ⟨ 𝐴 , 𝐵 ⟩ ↔ 𝐴 Btwn ⟨ 𝐵 , 𝐴 ⟩ ) )
7	3 4 4 5 6	syl13anc	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐴 Btwn ⟨ 𝐴 , 𝐵 ⟩ ↔ 𝐴 Btwn ⟨ 𝐵 , 𝐴 ⟩ ) )
8	2 7	mpbird	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐴 Btwn ⟨ 𝐴 , 𝐵 ⟩ )