Theorem elicc2i

Description: Inference for membership in a closed interval. (Contributed by Scott Fenton, 3-Jun-2013)

	Ref	Expression
Hypotheses	elicc2i.1	⊢ 𝐴 ∈ ℝ
	elicc2i.2	⊢ 𝐵 ∈ ℝ
Assertion	elicc2i	⊢ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elicc2i.1	⊢ 𝐴 ∈ ℝ
2		elicc2i.2	⊢ 𝐵 ∈ ℝ
3		elicc2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )