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Metamath Proof Explorer

Theorem iocleubd

Description: An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021)

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

Proof

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