Theorem iocleubd

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

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

Proof

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