Theorem icogelbd

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

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

Proof

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