Theorem iccleubd

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

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

Proof

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