Theorem iccgelb

Description: An element of a closed interval is more than or equal to its lower bound. (Contributed by Thierry Arnoux, 23-Dec-2016)

	Ref	Expression
Assertion	iccgelb	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		elicc1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
2	1	biimpa	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
3	2	simp2d	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝐶 )
4	3	3impa	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝐶 )