Theorem icoltubd

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

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

Proof

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