Theorem ioogtlb

Description: An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	ioogtlb	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) → 𝐴 < 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		elioo2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
2		simp2	⊢ ( ( 𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) → 𝐴 < 𝐶 )
3	1 2	biimtrdi	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) → 𝐴 < 𝐶 ) )
4	3	3impia	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) → 𝐴 < 𝐶 )