Theorem iooltubd

Description: An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 26-Jun-2021)

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

Proof

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