Theorem icoltubd

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

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

Proof

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