Theorem icogelbd

Description: An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	icogelbd.1	$$⊢\phi \to A\in {ℝ}^{*}$$
	icogelbd.2	$$⊢\phi \to B\in {ℝ}^{*}$$
	icogelbd.3	$$⊢\phi \to C\in \left[A,B\right)$$
Assertion	icogelbd	$$⊢\phi \to A\le C$$

Proof

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