Metamath Proof Explorer

Theorem icogelb

Description: An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019)

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

Proof

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		simp2	$⊢ (C \in ℝ^{*} \land A \leq C \land C < B) \to A \leq C$
3	1 2	biimtrdi	$⊢ (A \in ℝ^{} \land B \in ℝ^{}) \to (C \in [A, B) \to A \leq C)$
4	3	3impia	$⊢ (A \in ℝ^{} \land B \in ℝ^{} \land C \in [A, B)) \to A \leq C$