Theorem iccleubd

Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	iccleubd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	iccleubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
	iccleubd.3	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )
Assertion	iccleubd	⊢ ( 𝜑 → 𝐶 ≤ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		iccleubd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		iccleubd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		iccleubd.3	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )
4		iccleub	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ≤ 𝐵 )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → 𝐶 ≤ 𝐵 )