Theorem eliccre

Description: A member of a closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019)

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

Proof

Step	Hyp	Ref	Expression
1		elicc2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
2	1	biimp3a	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) )
3	2	simp1d	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ∈ ℝ )