Theorem ioogtlbd

Description: An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021)

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

Proof

Step	Hyp	Ref	Expression
1		ioogtlbd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		ioogtlbd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		ioogtlbd.3	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 (,) 𝐵 ) )
4		ioogtlb	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) → 𝐴 < 𝐶 )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → 𝐴 < 𝐶 )