icossre

Description: A closed-below interval with real lower bound is a set of reals. (Contributed by Mario Carneiro, 14-Jun-2014)

		Ref	Expression
	Assertion	icossre	$⊢ (A \in ℝ \land B \in ℝ^{*}) \to [A, B) \subseteq ℝ$

Step	Hyp	Ref	Expression
1		elico2	$⊢ (A \in ℝ \land B \in ℝ^{*}) \to (x \in [A, B) \leftrightarrow (x \in ℝ \land A \leq x \land x < B))$
2	1	biimp3a	$⊢ (A \in ℝ \land B \in ℝ^{*} \land x \in [A, B)) \to (x \in ℝ \land A \leq x \land x < B)$
3	2	simp1d	$⊢ (A \in ℝ \land B \in ℝ^{*} \land x \in [A, B)) \to x \in ℝ$
4	3	3expia	$⊢ (A \in ℝ \land B \in ℝ^{*}) \to (x \in [A, B) \to x \in ℝ)$
5	4	ssrdv	$⊢ (A \in ℝ \land B \in ℝ^{*}) \to [A, B) \subseteq ℝ$

Metamath Proof Explorer