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	$$⊢\phi \to A\in {ℝ}^{*}$$
	ioogtlbd.2	$$⊢\phi \to B\in {ℝ}^{*}$$
	ioogtlbd.3	$$⊢\phi \to C\in \left(A,B\right)$$
Assertion	ioogtlbd	$$⊢\phi \to A<C$$

Proof

Step	Hyp	Ref	Expression
1		ioogtlbd.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		ioogtlbd.2	$$⊢\phi \to B\in {ℝ}^{*}$$
3		ioogtlbd.3	$$⊢\phi \to C\in \left(A,B\right)$$
4		ioogtlb	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in \left(A,B\right)\right)\to A<C$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A<C$$