Theorem iccgelbd

Description: An element of a closed interval is more than or equal to its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021)

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

Proof

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