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

Proof

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