Theorem iocgtlbd

Description: An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022)

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

Proof

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