Metamath Proof Explorer

Theorem icoltub

Description: An element of a left-closed right-open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019)

		Ref	Expression
	Assertion	icoltub	$⊢ (A \in ℝ^{} \land B \in ℝ^{} \land C \in [A, B)) \to C < B$

Step	Hyp	Ref	Expression
1		elico1	$⊢ (A \in ℝ^{} \land B \in ℝ^{}) \to (C \in [A, B) \leftrightarrow (C \in ℝ^{*} \land A \leq C \land C < B))$
2		simp3	$⊢ (C \in ℝ^{*} \land A \leq C \land C < B) \to C < B$
3	1 2	biimtrdi	$⊢ (A \in ℝ^{} \land B \in ℝ^{}) \to (C \in [A, B) \to C < B)$
4	3	3impia	$⊢ (A \in ℝ^{} \land B \in ℝ^{} \land C \in [A, B)) \to C < B$